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Name _________________________________________ Geometry Common Core Regents Review 2018 TOPICS 1 – Trigonometry: 2 - Right Triangle Proportions and Angle Bisectors 3 - Transformations/Rigid Motions

TOPIC 1 - Trigonometry 1. Jami is building a roof for a dog house for her dogs, Cassie and Emmy. The dog house

pictured to the right with CAB CBA . The lower edge of the roof, located at AB, is 4

feet above the ground and the peak of the roof, located at C, is 6 feet above the ground. If

the horizontal distance from the lower edge of the roof to the peak of the roof is 5 feet, what is

the angle of inclination of the roof, to the nearest degree?

1) 22

2) 24

3) 50

4) 66

2. Scott, whose eye level is 1.5 m above the ground, stands 30 m from a tree. The angle of elevation as he looks to the

top of the tree is 36˚. How tall, to the nearest tenth of a meter, is the tree?

1) 21.8 m 2) 23.3 m 3) 19.1 m 4) 17.6 m

3.Right triangle WXY is similar to triangle DEF. The following are measurements in right triangle DEF: 𝑚∠𝐹 = 90°,𝐷𝐸 =√113, 𝐷𝐹 = 7, 𝐸𝐹 = 8. Which expression represents 𝑐𝑜𝑠𝑊?

1) 𝑐𝑜𝑠𝑊 =7

√113

2) 𝑐𝑜𝑠𝑊 =8

√113

3) 𝑐𝑜𝑠𝑊 = 7√113

4) 𝑐𝑜𝑠𝑊 = 8√113

4. In this figure, triangle GHJ is similar to triangle PQR. Based on this information, which ratio represents sin H?

1) 8

15

2) 8

17

3) 15

8

4) 17

8

2

5. In the figure below, a pole has two wires attached to it, one on each side, forming two right triangles. The wires form a right angle at the top of the pole.

a) How tall is the pole (nearest tenth of a foot)? b) How far from the base of the pole does Wire 2 attach to the ground? (nearest tenth of a foot) c) How long is Wire 2? (nearest foot) 6. Standing on the gallery of a lighthouse (the deck at the top of a lighthouse), a person spots a ship at an angle of depression of 20˚. The lighthouse is 28 m tall and sits on a cliff 45 m tall as measured from sea level. What is the horizontal distance, to the nearest meter, between the lighthouse and the ship?

7. Mariela is standing in a building and looking out of a window at a tree. The tree is 20 feet away from Mariela. Mariela’s line of sight to the top of the tree creates a 42° angle of elevation, and her line of sight to the base of the tree creates a 31° angle of depression. What is the height, in feet, of the tree?

8. Find 𝜃

sin cos 38

9. Find 𝜃

sin cos 3 20

10. Find 𝜃

sin 10 cos

3

11.The measure of an angle in a right triangle is x, and 𝑠𝑖𝑛𝑥 =1

3. Which of these expressions are also equal to

1

3?

1) cos(𝑥) 2) cos(𝑥 − 45°) 3) cos(45° − 𝑥) 4) cos(90° − 𝑥)

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TOPIC 2 – Right Triangle Proportions (HLLS and SAAS) 1. Given an equilateral triangle with sides of length 9, find the length of the altitude. Confirm your answer with the use of the Pythagorean Theorem.

2. A contractor designed a playground layout for a family’s backyard. Why is this layout not possible? Explain

10. In triangle ABC below, AC = 10, BC = 8, 𝑚∠𝐵 = 90°, and 𝑚∠𝐵𝐷𝐴 = 90°. How long is 𝐶𝐷̅̅ ̅̅ ? 1) 3.6 2) 5 3) 6.4 4) 4

11. In right triangle JKL, altitude KH is drawn to hypotenuse JL. If JH = 5 and JL = 20, find KL in simplest radical form.

12. Solve for x.

13. In the accompanying diagram, ∆𝑅𝑆𝑇 is a right triangle,

𝑆𝑈̅̅̅̅ is the altitude to hypotenuse 𝑅𝑇̅̅ ̅̅ . 𝑅𝑇̅̅ ̅̅ = 16, 𝑎𝑛𝑑𝑅𝑈̅̅ ̅̅ = 7. 𝑊ℎ𝑎𝑡𝑖𝑠𝑡ℎ𝑒𝑙𝑒𝑛𝑔𝑡ℎ𝑜𝑓𝑆𝑇?̅̅ ̅̅ ̅

a) 3√7 b) 4√7 c) 9 d) 12

SAAS HLLS 𝑆𝑖𝑑𝑒

𝐴𝑙𝑡𝑖𝑡𝑢𝑑𝑒=

𝐴𝑙𝑡𝑖𝑡𝑢𝑑𝑒

𝑆𝑖𝑑𝑒

𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

𝐿𝑒𝑔=

𝐿𝑒𝑔

𝑆𝑖𝑑𝑒

9

4

14. In the diagram below, the length of the legs AC and BC of right triangle ABC are 6 cm and 8 cm, respectively. Altitude CD is drawn to the hypotenuse of ΔABC

What is the length of AD to the nearest tenth of a centimeter? a) 3.6 b) 6.4 c) 6.0 d) 4.0

19. In the diagram below of , is a midsegment of , , , and . Find the perimeter of .

20. Triangle ABC is shown in the diagram below.

If joins the midpoints of and , which statement is not true?

1)

2)

3)

4)

21. In the diagram of below, A is the midpoint of B is the midpoint of , C is the midpoint of , and

and are drawn. If and , what is the length of ?

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TOPIC 3 – Transformations and Rigid Motions

Equilateral

Triangle

Square

Regular Pentagon

Regular Hexagon

# of sides 3 4 5 6 Angles of Rotation 360o/3 = 120o 360o/4 = 90o 360o/5 = 72o 360o/6 = 60o

1. A regular pentagon is rotated clockwise around

its center. The minimum number of degrees it must be rotated to carry the pentagon onto itself is:

1) 54° 2) 72° 3) 108° 4) 360°

2. Which regular polygon has a minimum rotation of 45° to carry the polygon onto itself? 1) Octagon 2) Decagon 3) Hexagon 4) Pentagon

3. Which of these symbols have rotational symmetry? A I only B IV only C I and III only D all except II

4. What are the coordinates of point under dilation ?

1) 2) 3) 4)

5. Point A is located at . The point is reflected in

the x-axis. Its image is located at

1. (-4, 7) 2. (-4, -7) 3. (4, 7) 4. (7, -4)

6. What is the image of point under a reflection

in the y-axis?

1. (3,1) 2. (-3,1) 3. (1,3) 4. (-1, -3)

Transformation: A change in the position, shape, or size of a figure.

Reflections

Rotations

Dilations

Dk (xk,yk)

Translations

Ta,b(x, y) = (x+a, y+b)

*Rotational Symmetry: A rotation that maps a figure back on to itself. *In regular polygons (polygons in which all sides are congruent) the number of rotational symmetries is equal to the number of sides of the figure.

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7. Triangle ABC has coordinates , , and

.

a. On the grid, graph and label . b. Graph and state the coordinates of , the

image of after a reflection over the x-axis c. Graph and state the coordinates of , the

image of after a translation to the left 10 units

d. Compare the area of to the area of e. Is congruent to ? Explain why or w hy not.

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3.Which equation represents circle O with center and radius 9?

1)

2)

3)

4)

4.What is the equation of a circle whose center is 4 units above the origin in the coordinate plane and whose radius is 6? 1)

2)

3)

4)

5. The coordinates of the endpoints of the diameter of a circle are and . What is the equation of

the circle? 1)

2)

3)

4)

6. The center of circle W is located at (-4, 2). Point (1,2) lies on this circle. Which point is also located on circle W? A (-7, -1) B (-4, 5) C (-1, -2) D (0, 7)

7. Identify the center and the radius of the following circle.

2 2 4 12 41x y x y

8.Identify the center and radius of the following circle.

2 210 6 9x x y y

Topic 5: Scale Drawings

1. AB has a length of 4 3 . What is the length of ' 'A B

after a dilation by a scale factor of 5?

1) 9 3 3) 9 8

2) 20 3 4) 20 15

2.Which of the following transformations preserves angle measure, but not distance?

1) Dilation 3) Reflection 2) Rotation 4)Translation

3.The perimeter of the smaller of two similar trapezoids is 18 units. The ratio of the sides of the smaller to the larger trapezoid is 3:5. What is the perimeter of the larger trapezoid?

1) 48 2) 30 3) 10.8 4) 6.75

4. Triangle ABC is similar to triangle DEF. Which is the correct statement for the ratio of their corresponding sides?

1) AB BC AC

DE DF EF

2) AB AC BC

DF DE EF

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3)

AB BC AC

DE EF DF

4)

AB BC AC

EF DF DE

5. In the following diagram, . If 3EC and

1(AD)

2BD , what is the length of AC ?

1) 6 2) 9 3) 12 4) 15

6. In the figure below, 𝑁𝑄̅̅ ̅̅ is parallel to 𝑂𝑃̅̅ ̅̅ and NQ=4cm, OP= 6cm, and MQ=8cm. If NO=2, how long is 𝑄𝑃̅̅ ̅̅ ? 1) 10 cm 2) 2 cm 3) 6 cm 4) 4 cm

7. Determine the scale factor of the given dilation from point O? 1) 2 : 3 2) 2 : 5 3) 3 : 2 4) 5 : 2

8. Which one of the following linear functions would remain unchanged under a dilation of 3 about the point (0, 3)?

1) 2y x

2) 2 3y x

3) 3 5 0y x

4) 3y x

9. ABC is similar to DEF . The ratio of the length of AB to the length of DE is 3:1. Which ratio is also equal to 3:1?

a) Dm

Am

b)

Fm

Bm

c)

DEFofarea

ABCofarea

d)

DEFofperimeter

ABCofperimeter

10.Marcos constructed the composition of dilations shown

below. Drawing 2 is 3

8 the size of Drawing 1, and Drawing 3

is twice the size of Drawing 2. What is the scale factor from Drawing 1 to Drawing 3?

11. Which similarity transformation maps ABC onto A’B’C’?

a) Dilation by factor of 2 and a Rotation about the point (0, 1)

b) Reflection over the origin and a Dilation by factor of 2

c) Rotation of 180 and a reflection over the line y=x

d) Translation (x+2, y+5) and a dilation of 2

2 cm3 cm

C

B

O

B'

C'

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12. Line segment AB with endpoints A(4, 16) and B(20, 4)

lies in the coordinate plane. The segment will be dilated

with a scale factor of 3

4 and a center at the origin to create

𝐴′𝐵′̅̅ ̅̅ ̅̅ . What will be the length of 𝐴′𝐵′̅̅ ̅̅ ̅̅ ?

1) 15

2) 12

3) 5

4) 4

13. PD is given with vertices P (a, b) and D (3a, 2b).

Find the length of the image of PD after a dilation with

scale factor of 3.

14. Triangle KLM is the pre-image

of Δ𝐾′𝐿′𝑀′, before a

transformation. Which statements

are true? Select all that apply.

a) Triangle KLM is similar to Δ𝐾′𝐿′𝑀′.

b) Triangle KLM is not similar to Δ𝐾′𝐿′𝑀′.

c) There was a dilation of scale factor of 0.5 centered at the origin.

d) There was a dilation of scale factor of 1 centered at the origin.

e) There was a dilation of scale factor of 1.5 centered at the origin.

f) There was a translation left 0.5 and up 1.5.

g) There was a translation left 1.5 and up 0.5.

15. Given ABC and its image ' ' 'A B C after a dilation with center at the origin.

a) Determine the constant of dilation and the ratio of A’B’:AB

b) Are these triangles congruent, similar or neither? Explain.

c) Find the ratio of the perimeter of Δ𝐴𝐵𝐶 to Δ𝐴′𝐵′𝐶′.

d) Find the ratio of the area of Δ𝐴𝐵𝐶 to Δ𝐴′𝐵′𝐶′

16. Triangles LMN and OPQ are shown below.

What additional information is sufficient to show that Δ𝐿𝑀𝑁 can be transformed and mapped onto Δ𝑂𝑃𝑄? 1) OQ = 6 2) MN = 9 3) ∠𝐿𝑀𝑁 ≅ ∠𝑄𝑂𝑃 4) ∠𝑁𝐿𝑀 ≅ ∠𝑄𝑂𝑃

17. Given Δ𝐴𝐵𝐶andΔ𝐷𝐸𝐹, 𝑚∠𝐴 = 56.3°,𝑚∠𝐵 =93.8°,𝑚∠𝐷 = 56.3°,𝑚∠𝐹 = 29.9°, 𝐴𝐵 = 3, 𝐴𝐶 =6, 𝐹𝐷 = 18, and𝐹𝐸 = 15. Find < 𝐸,< 𝐶, 𝐷𝐸,̅̅ ̅̅ ̅ &𝐵𝐶̅̅ ̅̅

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18. The center of dilation is: a) Greater than 1 b) Less than 1 c) Equal to 1 d) There is not enough information to answer this

19. Using the diagram (to the right) of the two similar triangles:

a. What is the relationship between the

sides of ∆𝐴𝐵𝐶 to the sides of ∆𝐴′𝐵′𝐶′?

b. Find the perimeters of both triangles. What is the

relationship between the perimeter of ∆𝐴𝐵𝐶 to the perimeter of ∆𝐴′𝐵′𝐶′?

c. Find the area of both triangles. What is the

relationship between the area of ∆𝐴𝐵𝐶 to the area of ∆𝐴′𝐵′𝐶′?

20. In the diagram below, XYZ is the result of a dilation of UVW . a) Find the values of 𝑋𝑍̅̅ ̅̅ and 𝑍𝑌̅̅̅̅ . b) Find the ratio of the sides.

c) Find the ratio of the perimeters. d)Find the ratio of the areas.

Topic 6 : Unknown Angles

Complementary Angles: Two angles that add to 90o

Supplementary Angles: Two angles that add to 180o

Adjacent Angles: Two angles that share a common vertex and common side.

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N

M

L

Right Angle An angle whose measure is 90o.

Angle Bisector A line, ray or segment which cuts an angle into two equal parts.

Perpendicular A line is perpendicular to another if it meets or crosses it at right angles.

Segment Bisector A line, ray or segment which cuts another line segment into two equal parts.

Equidistant A point P is equidistant from others if it is the same distance from them.

Linear Pair Two angles that are adjacent (share a leg) and supplementary (add up to 180°)

Straight Angle An angle whose measure is exactly 180° - a straight line

Midpoint A point on a line segment that divides it into two equal parts.The halfway point of a line segment.

Parallel Lines Lines are parallel if they lie in the same plane, and are the

same distance apart over their entire

length. (Never intersect)

Isosceles Triangle A triangle which has at least two of its sides equal in length

Vertical Angles

A pair of non-adjacent angles formed by the

intersection of two straight lines

Equilateral Triangle A triangle which has all three

of its sides equal in length.

Auxiliary Lines A line that us drawn

in to help demonstrate the

solution on a diagram

Corresponding Angles

MA B MA B

C

DA B

C

DA B

A

B

C

D

J

E

F

H

G

ao bo

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Alternate Interior Angles

Alternate Exterior Angles

Same Side Interior Angles

1.Find the measure of each labeled angle. Give a reason for your solution

2.Find the measures of angles a, b, c, d, and e.

3.In the diagram below of quadrilateral ABCD with diagonal BD, determine if AB is parallel to DC if

93,m ADB 43,m A 𝑚∠𝐶 = 3𝑥 + 5,𝑚∠𝐵𝐷𝐶 =

𝑥 + 19, and𝑚∠𝐷𝐵𝐶 = 2𝑥 + 6. Explain your reasoning.

4.In the diagram below of triangle HQP, side HP is

extended through P to T. a) Find 𝑚∠𝑄𝑃𝑇.

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b) Classify triangle PQH.

Use auxiliary lines to find the measures of angles 5-8. 5.

6.

7.

8.

Topic 7: Proofs/Congruence

Triangle Proofs Summary List the 5 ways of proving triangles congruent:

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1. SAS @ SAS 2. SSS @ SSS 3. AAS @ AAS 4. ASA @ ASA 5. HL @HL

Which two sets of criteria CANNOT be used to probe triangles congruent. 1. AAA@ AAA 2. SSA@ SSA

In order to prove a pair of corresponding sides or angles are congruent, what must you do first? Show the triangles are congruent first!

Property Meaning Geometry Example

Reflexive Property A quantity is equal to itself. 𝐴𝐵 = 𝐴𝐵

Transitive Property If two quantities are equal to the same quantity, then they are equal to each other.

If 𝐴𝐵 = 𝐵𝐶 and 𝐵𝐶 = 𝐸𝐹, then 𝐴𝐵 = 𝐸𝐹.

Symmetric Property If a quantity is equal to a second quantity, then the second quantity is equal to the first.

If 𝑂𝐴 = 𝐴𝐵, then 𝐴𝐵 = 𝑂𝐴.

Addition Property of Equality If equal quantities are added to equal quantities, then the sums are equal.

If 𝐴𝐵 = 𝐷𝐹 and 𝐵𝐶 = 𝐶𝐷, then 𝐴𝐵 + 𝐵𝐶 = 𝐷𝐹 + 𝐶𝐷.

Subtraction Property of Equality

If equal quantities are subtracted from equal quantities, the differences are equal.

If 𝐴𝐵 + 𝐵𝐶 = 𝐶𝐷 + 𝐷𝐸 and 𝐵𝐶 = 𝐷𝐸, then 𝐴𝐵 = 𝐶𝐷.

Multiplication Property of Equality

If equal quantities are multiplied by equal quantities, then the products are equal.

If m∠𝐴𝐵𝐶 = m∠𝑋𝑌𝑍, then 2(m∠𝐴𝐵𝐶) = 2(m∠𝑋𝑌𝑍).

Division Property of Equality If equal quantities are divided by equal quantities, then the quotients are equal.

If 𝐴𝐵 = 𝑋𝑌, then 𝐴𝐵

2=

𝑋𝑌

2.

Substitution Property of Equality

A quantity may be substituted for its equal.

If 𝐷𝐸 + 𝐶𝐷 = 𝐶𝐸 and 𝐶𝐷 = 𝐴𝐵,

then 𝐷𝐸 + 𝐴𝐵 = 𝐶𝐸.

Partition Property (includes “Angle Addition Postulate,” “Segments add,” “Betweenness of Points,” etc.)

A whole is equal to the sum of its parts.

If point 𝐶 is on 𝐴𝐵̅̅ ̅̅ , then 𝐴𝐶 + 𝐶𝐵 = 𝐴𝐵.

Statements of Equality

Picture Given Statement Reason

1.

CDAB <ADB and <BDC

are right angles

<ADB <BDC

Perpendicular Lines form right

Angles

All right <’s are

C

DA B

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N

M

L

N

M

L

2. EF intersects GH at J <EJH <GJF

Intersecting lines form

vertical <’s

3 EF bisects GH GJ JH

Bisector cuts segment in 1/2

4.

BD bisects <ABC <ABD <DBC Bisector cuts angle in 1/2

5.

M is the midpoint of AB

AM MB midpoint cuts segment in 1/2

6.

AB is a straight line <ADB & <CDB are

supplementary

Linear Pairs are supplementary

Picture Given Statement Reason

7.

∆LMN is

isosceles LN

is the base

ML ≅ MN

<MLN ≅ <MNL

Isosceles ∆ 2 sides are congruent and

base angles are congruent

8. LM ≅ MN <MLN ≅ <MNL When 2 sides are congruent the angles

opposite those sides are also

congruent

Converse also true if two angles are ≅

the sides opposite them are congruent

9. <1 <1 Reflexive

10.

<1 <3

<ABE <CBD

<2 <2

<ABE <CBD

<2 <2

<ABE <CBD

Reflexive

Addition

Reflexive

Subtraction

MA B

1

B

AC

1 32

B

ACD E

J

E

F

H

G

J

E

F

H

G

C

DA B

A

B

C

D

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A B

C D

E F

11.

AB CD

EF CD

AB EF Transitive

12.

m<1 + m<2 = 180

m<3 + m<4 = 180

m<1 + m<2 = m<3 +m<4

2 adjacent angles on a line = 180

2 adjacent angles on a line = 180

Substitution

1. For each pair of triangles, tell which triangles can be proven congruent (if at all) and by what method. A B C D

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2.Complete the missing parts of the flow chart proof to prove that if 𝑃𝑅̅̅ ̅̅ and 𝑄𝑆̅̅̅̅ bisect each other at T, then ∠𝑃 ≅ ∠𝑅.

21

43

17

3. Given: 𝐴𝐶̅̅ ̅̅ is the median to 𝐵𝐸̅̅ ̅̅ , 𝐸𝐶̅̅ ̅̅ is the median to 𝐴𝐷̅̅ ̅̅ . Prove: 𝐴𝐵̅̅ ̅̅ ∥ 𝐸𝐷̅̅ ̅̅

4. In the diagram of △ 𝑂𝑀𝑃 and △ 𝑂𝑄𝑁, ∠𝑀 ≅ ∠𝑄 and 𝑀𝑂̅̅ ̅̅ ̅ ≅ 𝑄𝑂̅̅ ̅̅ . Prove: 𝑀𝑁̅̅ ̅̅ ̅ ≅ 𝑄𝑃̅̅ ̅̅ .

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Ways to prove triangles are similar - Symbol ~ 5. Fill in the missing blanks with the correct reason for the proof.

6. Given: Quadrilateral ABCD diagonals AC and BD intersect at E, AC bisects BD, and ∠𝐷𝐴𝐸 ≅ ∠𝐵𝐶𝐸. Prove: 𝐴𝐵𝐶𝐷is a parallelogram

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7.

Given:

Prove:

PRT QRS

PR TR

QR SR

PR SR QR TR

8. Given: ABCD is a parallelogram Prove: AF = AB

STATEMENT REASON 1. Given

2. 𝑚∠𝐵 = 𝑚∠𝐴𝐷𝐶

3. 𝑚∠𝐴𝐹𝐶 = 𝑚∠𝐴𝐷𝐶

4.

5.

6.

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9. Given:

1 2

3 4

Prove:

AC BD

10. Given:

ABC DCB

EB EC

DB bisects ABC

AC bisects DCB

Prove:

BEA CED

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11. Given:

AB AC

RB RC

Prove:

SB SC

12. Given:

JK JL

JK XY

Prove:

XY XL

STATEMENTS REASONS

1) JK JL

JK XY

1) Given

2) 2) Isosceles triangles have 2 legs

3) 3) Base angles of an isosceles triangles are

4) 4) lines form corresponding angles

5) 5) Substitution

6) 6) Isosceles triangles have 2 base angles

7) 7) Legs of an isosceles triangle are

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Topic 8: Proving Properties of Geometric Figures Parallelogram Rectangle Rhombus Square

Opposite angles Consecutive angles supplementary Opposite sides Opposite sides parallel Diagonals bisect each other Diagonals bisect angles Diagonals to each other Diagonals Equiangular Equilateral

1. All of the following must have congruent diagonals

except a. A rectangle b. A square c. A parallelogram d. Isosceles trapezoid

2. A parallelogram must be a rhombus if the a. Diagonals are perpendicular b. Opposite angles are congruent c. Diagonals are congruent d. Opposite sides are congruent

3.If the measures of two opposite angles of a parallelogram are represented by 3x + 40 and x + 50, what is the measure of each angle of the parallelogram?

4.In parallelogram ABCD angle A can be represented by 3x + 20 and angle B can be represented by 7x – 40. Find the measure of each angle of the parallelogram.

5.In the diagram to the right, diagonals of rhombus ABCD (not drawn to scale) intersect at E.

105 xADEm and 42 xDAEm . 16AC and 12BD . a)Find DAEm and m DCB .

b)Find CD 6. In rectangle PQRS, if PQ = 3x – 8, QR = x + 12, and RS = 7. If the diagonals of a rhombus measure 12 and

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2x – 3, what is the value of SP?

16, what is the measure of a side of the rhombus?

a) 5 b) 10 c) 10√3 d) 20

8. The lengths of the diagonals of a rhombus are 8 centimeters and 12 centimeters. Find the perimeter of the rhombus.

9.In the accompanying diagram of parallelogram ABCD, diagonals AC and DB intersect at E. AE = 4x-15 and EC = 2x+1. Find AC.

10. In rhombus ABCD below, what is the measure of angle BDA?

11.Which statement is false?

1) All parallelograms are quadrilaterals. 2) All rectangles are parallelograms. 3) All squares are rhombuses. 4) All rectangles are squares.

12. The figure shows parallelogram ABCD with AE = 16. Let 𝐵𝐸 = 𝑥2 − 48 and 𝐷𝐸 = 2𝑥.What are the lengths of 𝐵𝐸̅̅ ̅̅ and 𝐷𝐸̅̅ ̅̅ ? Justify your answer.

13.The following represents a cyclic quadrilaterals. Find 14.The following represents a cyclic quadrilaterals. Find

24

the values of all variables shown.

the values of all variables shown

15. 16.

Topic 9: Solid Geometry

25

Not given on formula sheet: B – indicates area of the base

Density: a measure of the amount of substance in an object Density = 𝑴𝒂𝒔𝒔

𝑽𝒐𝒍𝒖𝒎𝒆

3D shapes

Oblique prism – bases are not aligned Polyhedron = a three-dimensional solid which consists of a collection of polygons,

Cross-section of a solid is the intersection through the solid parallel to the base

Slice of a solid is the intersection through the solid not parallel to the base (May also be referred to as a cross section)

When rotating 2D figure about a line what 3D figure will result

26

Triangle becomes cone rectangle becomes cyclinder Circle becomes a donut 1)

2)

3)

4)An isosceles right triangle is placed on a coordinate grid. One of its legs is on the x-axis and the other on the y-axis. Which describes the shape created when the triangle is rotated about the x-axis?

a) cone b) cylinder c) pyramid d) sphere

5)Calculate the area of the shaded figure below.

6)Two triangles △ 𝐴𝐵𝐶 and △ 𝐷𝐸𝐹 are shown below. The two triangles overlap forming△ 𝐷𝐺𝐶. The base of figure

𝐴𝐵𝐺𝐸𝐹 is comprised of segments of the following lengths: 𝐴𝐷 = 4,𝐷𝐶 = 3, and 𝐶𝐹 = 2. Calculate the area of the figure 𝐴𝐵𝐺𝐸𝐹.

7.Find the area of the shaded region:

a. triangle b. Square c. Rectangle d. pentagon

a. triangle b. Square c. Rectangle d. pentagon

27

8.Find the area of the shaded region:

9.Find the area of the shaded region:

10.This figure consists of 2 concentric circles. If the shaded area is 64𝜋𝑖𝑛2 and the smaller circle has a radius of 6 in., what is the radius, in inches, of the larger circle?

11. Find the area of the shaded region (round to the nearest hundredth).

Surface Area & Lateral Area

28

1) Find the surface area of a cube if the edge of the cube measures 6.4 centimeters.

2) If the surface area of a cube is 144 inches squared, find the length of the edge of the cube to the nearest tenth.

3)

a) Find the lateral area.

b) Find the surface area.

4) A a rectangular prism is 2 inches wide, 5 inches tall, and 12 inches long. a. Find the lateral area.

b. Find the surface area.

5)Given a square pyramid with an edge of 2 meters and a slant height of 8 meters, a. Find the lateral area

b. Find the surface area. 6)Find the lateral area of the following triangular pyramid

7) The nose cone of the rocket shown below is to be painted with a special sealant. How much surface area

8) A water tank is formed by a cylinder at its base and a hemisphere at the top. The radius of the cylindrical

29

needs to be covered with the sealant? (leave your answers in terms of )

base is 10 feet and the height of the cylinder is 30 feet. The exposed surfaces of the tank are being painted. If the paint being used covers 200 square feet per gallon, what is the minimum number of gallons needed to complete the job?

VOLUME 1)The volume of a rectangular prism is 144 cubic inches. The height of the prism is 8 inches. Which measurements, in inches, could be the dimensions of the base?

1) 3.3 by 5.5 2) 2.5 by 7.2 3) 12 by 8 4) 9 by 9

2) A paint can is in the shape of a right circular cylinder. The volume of the paint can is 600π cubic inches and its altitude is 12 inches. Find the radius, in inches, of the base of the paint can. Express the answer in simplest radical form.

3)The diameter of a hemisphere is 15 inches. What is the volume of the hemisphere, to the nearest tenth of a cubic inch?

4)A sphere is inscribed inside a cube with edges of 6 cm. In cubic centimeters, what is the volume of the sphere, in terms of π?

1) 12π 2) 36π 3) 48π 4) 288π

5)A cylinder has a height of 7 cm and a base with a diameter of 10 cm. Determine the volume, in cubic centimeters, of the cylinder in terms of π.

6)The volume of a sphere is approximately 44.6022 cubic centimeters. What is the radius of the sphere, to the nearest tenth of a centimeter?

1) 2.2 2) 3.3 3) 4.4 4) 4.7

7)As shown in the diagram below, a right pyramid has a square base, ABCD, and is the slant height. Which statement is not true?

1) 2) 3) 4)

∆𝐶𝐸𝐷 is isosceles

8)Find the volume of a cone with an altitude of 12 and a base with a diameter of 18. Express your answer in terms of π.

9)A rectangular prism has a base with a length of 25, a width of 9, and a height of 12. A second prism has a square base with a side of 15. If the volumes of the two prisms are equal, what is the height of the second prism?

30

1) 6 2) 8 3) 12 4) 15

10)Two prisms have equal heights and equal volumes. The base of one is a pentagon and the base of the other is a square. If the area of the pentagonal base is 36 square inches, how many inches are in the length of each side of the square base?

1) 6 2) 9 3) 24 4) 36

11)A right rectangular prism has a square base with an area of 12 square meters. The volume of the prism is 84 cubic meters. Determine and state the height of the prism, in meters.

12) A bulk cereal producer wants to fill a jumbo size box with its brand of fruit loops. They decide to fill the box 1 inch from the top instead of filling it completely. How many cubic inches of fruit loops will the producer save by not filling the box to the top?

13) Express the volume of the cube below, with a diameter d=6 feet.

14) In a solid hemisphere, a cone is removed as shown. Calculate the volume of the resulting solid. In addition to your solution, explain the strategy you used in your solution.

15) Eight bocce balls are in a box 18 inches long, 9 inches wide, and 4.5 inches deep. If each ball has a diameter of 4.5 inches, what is the volume of the space around the balls? Round to the nearest tenth of a cubed inch.

17) Find, to the nearest hundredth, the volume of the composite solid.

31

1)

2)The lateral faces of a regular pyramid are composed of: 1) Squares 2) rectangles 3)congruent right triangles 4)congruent isosceles triangles

3)Jennifer is trying to determine is there is any difference in shape for various horizontal cross sections of a cone. She works with two cones that are similar to the one pictured below. Jennifer takes a cross section of the first cone that is parallel to the base of the cone. Next she takes a cross section of the second cone that is slanted 5 degrees from parallel. What were her results?

1) The cross sections were both circles 2) The cross sections were both ellipses. 3) The first cross section was a circle and the second cross section was an ellipse. 4) The first cross section was an ellipse and the second cross section was a circle

4) A big theater in New York has 1.3 million square feet. At the last show, a band named SuperKron surveyed the

Things to remember PRINCIPLE OF PARALLEL SLICES IN THE PLANE: If two planar figures of equal altitudes have identical cross-

sectional lengths at each height, then the regions of the figures have the same area.

CAVALIERI’S PRINCIPLE: Given two solids that are included between two parallel planes, if every plane parallel to the two planes intersects both solids in cross-sections of equal areas then the volumes of the two solids are equal.

32

theater to make sure there wasn’t overcrowding. They found the population density to be 0.015 people per square foot. This included all people in the building. How many total people attended the theater on that day? 5) Trees that are cut down and stripped of their branches for timber are approximately cylindrical. A timber company specializes in a certain type of tree that has a typical diameter of 0.50 meters and a typical height of about 10 meters. The density of the wood is 380 kg per cubic meter and the wood can be sold by mass at a rate of $4.75 per kilogram. Determine and state the minimum number of whole trees that must be sold to raise at least $50,000.

TOPIC 10: Equation of a line 1. State whether these lines are parallel, perpendicular or neither:

𝑦 =1

3𝑥 − 10

𝑦 = 3𝑥

2. State whether these lines are parallel, perpendicular or neither: 𝑦 = 3𝑥 + 2 2𝑦 = 6𝑥 − 6

3. What is the slope of a line perpendicular to the line whose equation is y = -2/3 x – 5? (1) -3/2 (2) -2/3 (3) 2/3 (4) 3/2

4. Which equation represents a line parallel to the x-axis? (1) x = 5 (2) y = 10 (3) x = 1/3 y (4) y = 5x + 17

5. Identify the slope in the equation: 𝑥 +3

4= 4𝑥 − 𝑥 + 𝑦

6. Which equation represents a line that is parallel to the y-axis and passes through the point (4, 3)? (1) x = 3 (2) x = 4 (3) y = 3 (4) y = 4

7. Write the equation of a line that goes through the points (6,3) and (-6,1).

Things to remember

*Slope formula 𝑦2−𝑦1

𝑥2−𝑥1 (This formula is on the formula sheet)

* slope-intercept form is 𝑦 = 𝑚𝑥 + 𝑏 where m = slope, b = y-intercept (where the line crosses the y-axis)

*point-slope form is: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1).

* Parallel lines have the same slopes. * Perpendicular lines have negative reciprocal slopes. * The normal line – is the line perpendicular to a given point/line * Circumcenter – is the point of concurrency of three perpendicular bisectors of each side of the triangle

33

8. Write equation of a line that goes through the origin and (4,-2). 9. Write the equation of the line that has a slope of 3/4 and passes through the point (4, 1) in point-slope form. 10. Write the equation of a line that is perpendicular to the line whose equation is y = 3/5 x - 2 and passes through the point (3, -6). Express your answer in slope-intercept form.

11. Which of the following represents an equation for the line that is parallel to 𝑦 =3

2(𝑥 − 6)and which

passes through the point (6,-8)?

a. 𝑦 − 8 = −2

3(𝑥 + 6) b. 𝑦 − 8 =

3

2(𝑥 + 6) c. 𝑦 + 8 =

3

2(𝑥 − 6) d. 𝑦 + 8 = −

2

3(𝑥 − 6).

12. Find the equation a line passing through the point (-9,-3) and perpendicular to the line x+2y+2 = 0.

13. Given S(−4,−1)and𝑇(7, 1):

a. Find the equation for the normal line b.Find the equation for the normal line

through 𝑆𝑇̅̅̅̅ at point S. through 𝑆𝑇̅̅̅̅ at point T.

14. Write an equation of the line that is the perpendicular bisector of the line segment having endpoints

34

(2, 5) and (-6,-1)

15. Triangle ABC has vertices A(0,0), B(6, 8) and C(8, 4). Which equation represents the perpendicular bisector of BC?

a) 𝑦 = 2𝑥 − 6

b) 𝑦 = −2𝑥 + 4

c) 𝑦 =1

2𝑥 +

5

2

d) 𝑦 = −1

2𝑥 +

19

2

16. The diagram below shows the construction of the center of the circle circumscribed about . This construction represents how to find the intersection of 1) the angle bisectors of 2) the medians to the sides of 3) the altitudes to the sides of 4) the perpendicular bisectors of the sides of

17. The vertices of ∆DEF are D (1,1), E (5,5), and F (-1,5). Find the coordinates of the circumcenter of the triangle

[the use of the grid is optional].

18. Describe the location of the point of concurrency of the perpendicular bisectors of a triangle.

35

a) on the longest side of the triangle

b) in the same place as the point of concurrency of the altitudes of the triangle

c) always in the interior of the triangle

d) in the exterior, on, or in the interior of the triangle

e) none of the above

19.Line y = 4x − 1 is transformed by a dilation with a scale factor of 2 and centered at (2,7). What is the line’s image?

a) y = 4x -7 b) y = 4x – 1

c) y = 8x – 1 d) y = 8x – 7

20.Line y = 2x + 6 is transformed by a dilation with a scale factor of 1

2 and centered at the origin. What is the line’s

image?

a) y = 2x + 6 b) y = x + 3 c) y = 2x + 3 d) y = x + 6

21.Line y = 1

3x − 1 is transformed by a dilation with a scale factor of 3 and centered at the origin. What is the line’s

image?

a) y = 1

3x − 1 b) y =

1

3x − 3

c) y = x − 1 d) y = x − 3

22.Use the diagram shown. Dilate line m about the origin with a scale factor of 2. What is the equation of the line’s image?

Topic 11: Perimeters and Areas of Polygonal Regions in the Cartesian Plane

1) a) Sketch the region bound by the following equations.

Things to remember Tips to finding area and perimeter on the coordinate plane

Horizontal lines are in the form y = #; vertical lines are in the form x = # Other types of lines can be drawn using either the slope-intercept method or with a table from

a graphing calculator (be sure to get the equation in y = mx + b form first!) Be sure to clearly find the points of intersection of your lines to aid in creating the triangle or

quadrilateral that will be formed by the given equations; a highlighter can help! For perimeter: if the shape has vertical and horizontal lines, count the units for side lengths; if

not, the distance formula 2 2

2 1 2 1( ) ( )d x x y y can be used to find diagonal side lengths

(watch simplest radical form) For area: if the shape has vertical and horizontal lengths simply plug those into the

appropriate formulas; if not, draw a rectangle or square around the shape and subtract out the areas of the outer pieces from the total area of the rectangle/square. Once again, a highlighter can help!

36

b) Determine the vertices of the triangle graphed. c) Prove or disprove that the triangle is isosceles. d) Find the area of the triangle.

3

8

2 6

y

x

y x

2) a) Sketch the region bound by the following equations. b) Determine the vertices of the quadrilateral graphed. c) Find the perimeter of the quadrilateral in simplest radical form. d) What type of quadrilateral is this? Explain.

6

2 2 12

8 4 4

2

y x

x y

x y

y x

TOPIC 12: Partitioning and Extending Segments

37

1. Find the coordinates of the point P that lies along the directed line segment from A(3,1) to B(6,7) and partitions the segment in the ratio 2 to 1.

2. Find the coordinates of point F on the directed segment from D to E that divides it into a ratio of 1 to 3.

3. Given If point S lies of the way along , closer to F than to G, find the coordinates of S.

4. The directed line segment from C(-4, 5) to D(12, 13) is partitioned by point P(-2,6). What is the ratio of CP to DP? 1) 1:1 3) 1:2 2) 1:4 4) 1:8

Topic 13: Coordinate Geometry

-2, 0( ) 5, 8( )

0, 2 and 2, 6 .F G 5

12FG

Things to remember Partitioning a Line Segment Scan for a Youtube clip on how to partition a line segment into a certain ratio.

38

Ex) Fill in the properties chart below.

A quadrilateral where

1. Opposite sides are ______________.

2.Opposite angles are _____________.

3.Diagonals _____________________________________.

4. Consecutive angles add to _____________.

5. One pair of opposite sides is both _____ and _____.

A parallelogram where 1. All angles are ______________. 2. Diagonals are ______________.

A parallelogram where 1. All sides are ______________. 2. Diagonals are ______________.

A parallelogram where

1. All sides are ______________.

2. All angles are ______________.

3. Diagonals are ______________ and ______________.

Things to remember Proving a Parallelogram To prove that a quadrilateral is a parallelogram, prove any one of the following statements is true:

Things to remember: Distance

2 2( ) ( )x x y y

Midpoint

,2 2

x x y y

Slope

y y

x x

39

1. Find the slope of a line that contains the points: (9,5)(6,-1)

2. A line segment with coordinates (3,2) and (-1,y) has a

slope of 5

2. Find the value of y.

3. Which statements describe properties of the diagonals of a rectangle?

4. Which statement is true? a) A quadrilateral is always a parallelogram

40

I The Diagonals are congruent II The diagonals are perpendicular III The diagonals bisect each other

a) II and III, only c) I and III, only

b) I and II, only d) I, II, and III

b) A square is always a parallelogram c) A parallelogram is never a rhombus d) A trapezoid never has two congruent sides

5. The coordinates of point R are (-3,2) and the coordinates of point T are (4,1). What is the length of RT?

a) 2√2 b) 5√2 c) 4√3 d) √10

6. The coordinates of the endpoints of PQ are P(3a,4b) and Q(2a,3b). The length of PQ must equal

a) a+b b) √𝑎2 + 𝑏2

c) 𝑎2 + 𝑏2 d) √25𝑎2 + 49𝑏2

7. The midpoint AB has coordinates of (-1,5). If the coordinates of A are (-3,2), what are the coordinates of B? 8) Quadrilateral ABCD has coordinates A(-1,1), B(4,5), C(9,1) and D (4,-3). Using coordinate geometry, prove that: a) ABCD is a rhombus b) ABCD is not a square (The use of the graph is optional)

9) The coordinates of Quadrilateral PQRS are P(-4,0), Q(0,1), R(4,-1) and S(-4,-3). Prove that: a) PQRS is a trapezoid

41

b) PQRS is not an isosceles trapezoid (The use of the graph is optional)

10) Quadrilateral KLMN has coordinates K(-2,3), L(4,6), M(3,2) and N(-3,-1). Using coordinate geometry, prove that the diagonals bisect each other

11) Given Triangle ABC, A(-1,2), B(7,0) , and C(1,-6) and a point D(4,-3) on BC. Using coordinate geometry. Prove that: a) AD is the perpendicular bisector of BC b) Triangle ABC is isosceles

Topic 14: Circles

Things to remember * Central angle = to the intercepted arc * Inscribed angle = half of the intercepted arc * Part x Part = Part x Part

42

1. In the accompanying diagram of circle O, the measure

of is 64º.

What is ? 1) 32 2) 64 3)96 4)128

2. In the accompanying diagram of circle O,

What is ? 1) 210 2)105 3)95 4)75

3. In the diagram below, is tangent to circle O, and

is a chord. If , find the measure of .

4. In the accompanying diagram, is tangent to circle O

at B, is a chord, and is a diameter. If

, find .

5. The accompanying diagram shows two lengths of

wire attached to a wheel, so that and are

tangent to the wheel. If the major arc has a measure of 220°, find the number of degrees in .

6. In the diagram below, circle O has a radius of 5, and

. Diameter is perpendicular to chord at E.

What is the length of ?

7. In the diagram below of circle O, chords and

intersect at E, , and .

8. In the diagram below, chords and intersect at E.

If , , and , what is the

43

What is the degree measure of ? 1) 87 2) 61 3) 43.5 4) 26

value of x?

1) 12 2) 20 3) 30 4) 60

9. In the diagram of circle O below, chord is parallel

to diameter and .

What is ? 1) 150 2) 120 3) 100 4) 60

10. The accompanying diagram represents circular pond O with docks located at points A and B. From a cabin located at C, two sightings are taken that determine an

angle of 30° for tangents and . What is ?

1) 30 2) 60 3) 75 4) 150

11.

12.

44

13. In a circle, chords AB and CD intersect at E. If AE=21, EB=5, and ED=7, find CE.

14.

15.

16. In the accompanying diagram of circle O, and are diameters. Which statement is not true?

1) 2) 3) 4)

17. In the accompanying diagram, is inscribed in

circle O and is a diameter.

What is the number of degrees in ? 1) 30 2) 45 3) 60 4) 90

18.

45

160°

𝒙

19. What is the measure of the arc if the sector has a

radius of 12 and a central angle which measures 6radians?

20.What is the area of the sector of a circle with a radius of

10cm, when the central angle of the sector is 60°, to the

nearest hundredth?

21.Circle B has a radius of 14 cm. Angle B intercepts the arc with a length of 6𝜋. Find the measure of angle B in radians to the nearest hundredth.

22.Find the exact area of the following annulus.

23.The sector area, shaded in the circle below, equals

16𝛑 square cm. What is the radius of the circle?

24.If the 𝑚 < 𝐵𝐴𝐶 = 54. What is the value of x?

25.Find the area of the shaded region to the nearest hundredth if <BAC = 62°

4

3

46

Topic 15: Basic Constructions

Construct an Equilateral Triangle

Copy an Angle

Bisect an Angle

Construct a Perpendicular Bisector

Construct a Perpendicular Bisector Construct a Perpendicular Bisector Through a point on the line Through a point not on the line

47

1. Use a compass & straightedge to construct ,3OD ADB

2. Use construction tools to create a scale drawing of

△ 𝑋𝑌𝑍 with a scale factor of 𝑟 =1

2 and center of dilation at

Z.

3. Construct a 30° − 60° − 90° triangle using only a compass and straightedge. Explain why your construction works. If the side opposite the 30° angle is represented by y, represent the other two sides of the triangle in terms of y. 4. Using constructions, find the center of the circle. Then, inscribe a square in the circle.

O

A

B

D

48

5. Construct a 45° − 45° − 90° triangle using only a compass and straightedge. If one leg of your triangle measures x, express the length of the hypotenuse in terms of x. 6. Perform the following dilation on circle A: 𝐷𝑂,2 [Hint: Draw radius 𝐴𝑃̅̅ ̅̅ and perform the dilation to create 𝐴’𝑃’̅̅ ̅̅ ̅]

8. Triangle FUN is shown below. Is it an equilateral triangle? Justify your answer.

49

9) Construct a line parallel to the given line and passing through the given point.

Topic 16: Transformations/Rigid Motions

Rigid Motion: A transformation that changes only the position of the figure(length and angle measure are preserved). Image: The result of a transformation of a figure(called the pre-image). To identify the image of a point, use the prime notation. The image of point A is A’ (read as A prime). Isometry: A transformation that does not change in size. These include all of the rigid motions: reflections translation and rotations.

Direct Isometry: Preserves size and order(orientation) of the vertices.

Opposite Isometry: Preserves the size, but the order of the vertices changes.

Type of Rigid Motion

Reflection ROTATION TRANSLATION

Is size preserved? YES YES YES Is orientation

preserved?

NO YES YES

Which type of Isometry?

OPPOSITE ISOMETRY DIRECT ISOMETRY DIRECT ISOMETRY

50

Rotation Reflection Translation

RIGID MOTIONS WITH THE COMPASS Rotations

Rotate 60o clockwise through a point. (Construct Equilateral Triangle)

Find the center of a rotation(Construct the perpendicular bisector for two pairs of corresponding points-

where these two intersect is the center of rotation)

Reflections

Reflect over a line(Construct perpendicular from A through L)

51

Find the line of Reflection(Construct Perpendicular bisector of any two corresponding points).

Translation

Translate along a given vector.

1. Determine the line of reflection for ABC and A’B’C’.

C'

A'

B'

B

A C

52

2. Use a compass and straight edge to construct ( )mR ABC (reflection over line m). Is orientation

preserved? Is distance preserved? Angle measures? Is it a rigid motion? Is it a direct isometry?

3. Jeff states that the following transformation is a rotation. He is correct. Explain why this is a rotation and

not a translation or a reflection using the characteristics of these transformations.

4. Determine point O, the center of rotation.

5. Without using a protractor, construct

O,60 ( )R ABC (the rotation of 60° about point O). Is

m

B'

A'

C'C

A

B

A'

B'

C'

C

B

A

53

orientation preserved? Is distance preserved? Angle measures? Is it a rigid motion? Is it a direct isometry?

6. Use a compass and straight edge to construct the following translation. Is orientation preserved? Is distance preserved? Angle measures? Is it a rigid motion? Is it a direct isometry?

Construct the inscribed polygons below. Use QR codes below if need be.

C

BA

O

54

1. Construct an inscribed hexagon. 2. Construct an inscribed equilateral triangle.

3.Construct an inscribed square 4. Construct an inscribed octagon (Hint 360/8 = 45 and 45 is half of 90

Topic 17: Concurrence

55

Points of Concurrency

Types of Segments What this type of line or segment does

Located Inside or Outside of the

Triangle

Circumcenter

Perpendicular

Bisectors

Forms a right angle and cuts a side in ½

*Center of a triangle’s circumcircle*

Both; depends on the type of triangle

Incenter

Angle Bisectors

Cuts an angle in ½

*Center of a triangle’s incircle*

Inside

Centroid

Medians

Cuts a side in ½

Inside

Orthocenter Altitudes Forms a right angle with the

side Both; depends on the

type of triangle Constructions: Circumcenter Incenter Centroid Orthocenter

CIRCUMCIRCLE INCIRCLE

Use the diagram to identify where each point of concurrency will lie.

56

Ex) The incenter will lie on

(1) AD (2) AE (3) AF (4) GF Ex) The circumcenter will lie on

(1) AD (2) AE (3) AF (4) GF Ex) The centroid will lie on

(1) AD (2) AE (3) AF (4) GF Ex) The orthocenter will lie on

(1) AD (2) AE (3) AF (4) GF

1.In shown below, P is the centroid and

. What is the length of and PF ?

2.In the diagram below, is a median of triangle

PQR and point C is the centroid of triangle PQR.

If and , determine and state the

length of .

3.In the diagram of below, medians

and intersect at point F. If , what is

the length of ?

4.In the diagram below of , ,

and .

Point P must be the

a) centroid b) circumcenter

c) incenter d) orthocenter

5. Triangle ABC has vertices , , and . Determine the point of intersection of

57

the medians, and state its coordinates. [The use of the set of axes below is optional.]

6.In the diagram of triangle ABC below, Ashley found centroid P by constructing the three medians.

She measured AD and found it to be 24 inches. Find AP and PD.

7.Find the centroid of a triangle whose vertices

are (-1,3 )(2,1) and (8, -4).

8.Find the coordinates of the centroid of triangle

XYZ with X(-4, 2) Y(0,5) and Z (3, -1)