HUSD High School Geometry Semester 2 Study Guide · 2013-05-15 · HUSD High School Geometry Semester 2 Study Guide Page 1 of 19 MCC@WCCUSD (HUSD) 03/09/13 1 ... The diagram shows

HUSD High School Geometry Semester 2 Study Guide

Page 1 of 19 MCC@WCCUSD (HUSD) 03/09/13

1 In the figure below,

€

sinA = 0.7.

€

cosA = 0.714

What is

€

AC and AB? Solutions:

€

sinA =oppositehypotenuse

0.7 =21AC

0.7 AC( ) = 21

AC =210.7

AC =210.7

⋅1010

AC =2107

AC = 30

€

cosA =adjacent

hypotenuse

cosA =AB30

0.714 =AB30

30 0.714( ) = AB21.42 = AB

Trig 12.0/G.SRT.6,7

1´ You try: 1) a) Find

€

AB and

€

BC . b) The diagram shows an 8-foot ladder leaning against a wall. The ladder makes a 53˚ angle with the wall. What is the distance up the wall the ladder reaches?



2 What is the height of the streetlight in the figure below? Round your answer to the nearest tenth. Solution:

€

tangent =oppositeadjacent

tan40 =h

20

20 tan40( ) =h

2020( )

16.8 = hh =16.8 ft.

21.0/G.C.2

3 What is the median of a triangle?

A median of a triangle is a line segment connecting a vertex to the midpoint of the opposite side.

1.0/G.CO.10

2´ You try: 2) What is the height of the tree in the figure below? Round your answer to the nearest tenth.

3´ You try:

3) Name a median

E

D

F

C

B

A



4

If mABC = 300° , what is m ACD∠ ?

If mABC = 300° , then

mAC = 360°−300°

= 60°

m∠ACD = 12⋅mAC

=12i60°

= 30°

21.0/G.C.2

5 Complete the following statement (of a

theorem): If two chords are equidistant from the center of a circle, then ______________________. Solution: The chords are congruent.

21.0/G.C.2

6 What can you say about opposite angles of an

inscribed quadrilateral? They are supplementary, or their measures add to 180° .

13.0/7.G.5

4´ You try: 4)

If BC

is tangent to circle O at B and 120m AOB∠ = ° , find m DBA∠ .

5´ You try:

5) Given: Circle O.

B A

D

C

O

AB ? CD

6´ You try:

6) What is the value of y?

A B

C D

3y°

4y°

A

B C

D

O

E .



7 For which quadrilaterals do the diagonals bisect each other? The diagonals bisect each other in all parallelograms, which include rectangles, rhombuses, and squares.

1.0

8 In quadrilateral ABCD, 4AB = , 8BC = ,

5CD = , and 9DA = . What are the possible values of BD? First, draw quadrilateral ABCD and put in the information. Using the Triangle Inequality Theorem, In ABD , BD has to be greater than 5 and less than 13. In BCD , BD has to be greater than 3 and less than 13. 5 13BD∴ < <

6.0

9 How do you find the length of the

midsegment (or median) of a trapezoid? Solution: You compute the average (add and divide by 2) of the lengths of the parallel sides/bases of the

trapezoid. 1 2Average2

b b+=

OR Midsegment is the value equidistant from 1 2b b+

7.0

7´ You try: 7a) For which polygons are opposite angles congruent? 7b) If DB bisects AC at E, what can you say about AE and EC?

8´ You try:

8a) What is the greatest possible value of AB? Why?

8b) In ABC , if 3AB = and 7AC = ,

describe the possible lengths of BC.

9´ You try:

9a) Define midsegment of a trapezoid. Use the following figure for 9b and 9c

9b) If 8AB = and 12DC = , what is EF? 9c) If 8EF = and 6AB = , what is DC?

B A

C D

4

8 9

5

B

A

C 8 5

1b

2b

D C

B A

E F



10 The sum of the measures of the interior angles of a polygon is 720°. How many sides does the polygon have? ( 2) 180 720

2 46

nnn

− ⋅ =− =

=

The polygon has 6 sides.

12.0

11 If ( 2) 180n− ⋅ is the sum of the measures of

the interior angles of any polygon, what is the measure of each interior angle of a regular polygon?

n stands for the number of sides, so the number of angles is also n. Therefore, the measure of each angle is the sum of the measures of all of the angles divided by the

number of angles, or ( 2) 180nn

− ⋅ .

12.0

10´ You try: 10a) How do you find the sum of the measures of the angles of a pentagon?

10b) What does the n in ( 2) 180n− ⋅ stand for?

11´ You try:

11) Each interior angle of a regular polygon has measure 120º. How many sides does the polygon have?



12 The apothem of a regular hexagon is 3 3 and the measure of each side is 6. What is the area of the hexagon? A regular hexagon is made up of 6 congruent triangles: The apothem is the height, h, of each triangle. The side length is the base, b, of each triangle. Area of the hexagon = (Area of 1triangle)(number of triangles)

= 12

bh⎛⎝⎜

⎞⎠⎟

6( )

= 12•6•3 3

⎛⎝⎜

⎞⎠⎟

6( )= 3•3 3( ) 6( )= 9 3 6( )= 54 3 units2

The area of the hexagon is 54 3 square units. OR Area of a regular polygon, where a = apothem and p = perimeter of the regular polygon

A= 12iaip

( )( )1 3 3 362

A =

A = 54 3 units2

10.0/7.G.6

12´ You try:

12) The apothem of a regular hexagon is

4 3 and the measure of each side is 8.

What is the area of the hexagon?



13 Find the apothem of a regular quadrilateral whose area is 10 square units. Area of square = Area of 4 triangles

A = 4iA

10 = 4 12bh

!

"#

$

%&

10 = 4 1210 ia

!

"#

$

%&

10 = 2 10 ia

102 10

=2 10 ia2 10

102 10

i 1010

= a

10 102i10

= a

102

= a

The apothem is 102

units.

10.0/7.G.6

13´ You try: 13) Find the apothem of a regular quadrilateral whose area is 12 square units.

x

x a

x

x a

x

x

x

x

Area = bh10 = xix10 = x2

10 = x

OR, for a square,

Apothem12x=

a = 12x

=12

10( )=102



14 The radius of a circle is 6 and its circumference is 12π . The area of the circle is:

A = πr2

= π 6( )2

= 36π

The area of the circle is 36π square units.

Note: the circumference was extraneous information.

10.0/6.G.1

15 For which quadrilaterals are each of the

following true:

a. the diagonals bisect each other. b. the diagonals are congruent.

c. the diagonals are perpendicular.

a. parallelograms, rectangles, rhombuses, and squares

b. rectangles, squares c. rhombuses, squares, and kites

1.0/G.CO.8

16 a. What is the sum of the measures of the

exterior angles of a polygon?

The sum of the measures of the exterior angles of a polygons is 360º.

b. What does it mean for a polygon to be a regular polygon?

All the sides are congruent and all the angles are congruent.

12.0/8.G.5

14´ You try: 14a) The radius of a circle is 9 and its

circumference is 18π . The area of the

circle is: 14b) The area of a circle is 54π . Find its radius in simplest form.

15´ You try:

15) True or false?

a. All rectangles are parallelograms. b. All parallelograms are rectangles.

c. All squares are rectangles. d. All squares are rhombuses.

16´ You try:

16) Find the measure of each exterior angle

of a regular heptagon.



17 A trapezoid has a base that measures 45 and another base that measures 15. Its area is 900. Find its height. Area of a Trapezoid = Avg. of the bases ih

OR ( )1 212

A h b b= +

900 = 45+152i h

900 = 602ih

900 = 30h90030

=30h30

30 = h

OR

The height of the trapezoid is 30 units

10.0/6.G.1

18 A square pyramid has an altitude with a

length of 15 cm and a base with sides measuring 10 cm. Find the volume of the pyramid. Volume of a pyramid is

1 ,3

V Bh= where B is the area of the base and

h is the height. The base of the pyramid is a square with side lengths of 10 cm.

V =13Bh

V =13s2( )h

V =1310cm( )

215cm( )

=100cm2( ) 3i5cm( )

3=100i5cm3

= 500cm3

9.0/8.G.9

17´ You try: 17) A trapezoid has a base that measures 43 and another base that measures 59. Its area is 1,020. Find its height.

18´ You try:

18) Find the volume of a right circular cylinder whose radius is 4 in and height is 7 in.

A= 12h b1 +b2( )

900 = 12h 45+15( )

900 = 1260( )h

900 = 30h30i3030

= h

30 = h



19 A regular pentagonal pyramid has a base whose edge is 4 and a slant height of 6. Find the lateral area.

First put in the information:

The lateral area is the sum of the areas of the lateral faces.

In a pentagonal pyramid, there are five triangular lateral faces. Because the pyramid is regular, all five lateral faces are congruent and look like this:

Area of one triangle

( )Lateral Area 5 1260

==

∴ The lateral area is 60 square units

9.0/7.G.6

19´ You try:

19) For a regular triangular pyramid, the edge of the base measures 6 and the

lateral edges measure 5. Find the lateral area.

4

6

A =12bh

=124( ) 6( )

=12

4

6



20 Find the lateral area of a right circular cylinder if the height is 10 and the radius is 3. Solution: The lateral area is the surface area of the side, which is shaped like a rectangle. Area of a rectangle = l · w = circumference (h)

( )( )( )

2 10

2 3 1060

rπππ

=

==

The lateral area is 60π square units. 9.0/G.GMD.3

End of Study Guide

20´ You try:

20) Find the lateral area of a right circular cylinder if the height is 8 and the base

has a circumference of 27.



You Try Solutions: 1´ You try:

1) a) Find

€

AB and

€

BC . For side

€

AB:

€

cos32 =10AB

AB( )cos32 =10AB

AB( )

AB( )cos32 =10

AB =10

cos32

AB =11.79

For side

€

BC :

€

sin32 =x

11.79

11.79( )sin32 =x

11.7911.79( )

11.79( )sin32 = x6.25 = x

1´ You try continued: 1) b) The diagram shows an 8-foot ladder leaning against a wall. The ladder makes a 53˚ angle with the wall. What is the distance up the wall the ladder reaches?

€

Let ? = x

cos53 =x8

8( )cos53 =x8

8( )

4.81= x

€

∴ the ladder is about 4.81 feet up the wall.



2´ You try: 2) What is the height of the tree in the figure below? Round your answer to the nearest tenth.

€

Let x = the height of the tree

€

sin50 =x100

100( )sin50 =x100

100( )

100( )sin50 = x76.6 = x

€

∴ the tree is about 76.6 feet tall.

3´ You try:

3) Name a median

CE is a median. AE EB≅ , so E is the midpoint of AB .

4´ You try: 4)

If BC

is tangent to circle O at B and 120m AOB∠ = ° , find m DBA∠ .

AOB∠ is a central angle intercepting AB .

Therefore, mAB =120°

m∠ABC =12mAB

= 60°

180(180 60)120

m DBA m ABCm DBA

∠ + ∠ = °∠ = − °

= °

OR

mAEB = 360−120( )°= 240°

m∠DBA= 12mAEB

=12(240°)

=120°

OR

€

m∠OBD = 90 radius to a point of

tangency is ⊥ to tangent

€

ΔAOB is an isosceles Δ∴m∠BOA = 30

€

m∠DBO = m∠OBA =120 Angle Sum Thm.

A

B C

D

O

E .

E

D

F

C

B

A

A

B C

D

O

E .

Triangle Sum Thm. Isosceles Triangle Thm.



5´ You try: 5) Given: Circle O.

B A

D

C

O

AB ? CD Distance is measured perpendicularly, so AB and CDare the same distance from the center, O. Therefore, AB CD≅ .

6´ You try:

6) What is the value of y? Opposite angles of a quadrilateral that is inscribed in a circle are supplementary. Therefore, OR, the measure of an inscribed angle is half of the measure of the intercepted arc, therefore,

€

3y + 4 y =1

2360( )

7y = 180

7y7

=1807

y =180

7

7´ You try: 7a) For which polygons are opposite angles congruent? Opposite angles are congruent for all parallelograms. Rectangles, rhombuses, and squares are all parallelograms, so opposite angles are also congruent for rectangles, rhombuses and squares. 7b) If DB bisects AC at E, what can you say about AE and EC?

andAE EC will be congruent (definition of bisect)

3y°

4y°

3 4 1807 1807 1807 7

1807

y yyy

y

+ ==

=

=

D

A

B

C

E



8´ You try: 8a) What is the greatest possible value of AB? Why? If AC and CBwere laid end-to-end, or connected in a straight line, then AB would look like this: And the length of ABwould be 13. In order to make a triangle, however, ACB∠ must bend; that is, its measure must be less than 180º, like this: this: or this: In each (and every) case, AB must be less than 13.

OR

By the Triangle Inequality Theorem,

5 813

AC CB ABABAB

+ >+ >

>

8´ You try continued:

8b) In ABC , if 3AB = and 7AC = ,

describe the possible lengths of BC. In ABC Possible lengths of BC

3 74

BCBC

+ >>

3 710

BCBC

+ >>

OR

7 3 7 34 10BCBC

− < < +< <

B

A

C 8 5

A

C B

5 8

A C B

5 8

A C B

5 8

A C

B

5 8

Large values

A

B

C

3

7

Small values

A

B

C 3 7



9´ You try:

9a) Define midsegment of a trapezoid. The midsegment of a trapezoid is the segment that connects the midpoints of the legs of a trapezoid. Use the following figure for 9b and 9c

9b) If 8AB = and 12DC = , what is EF? 9c) If 8EF = and 6AB = , what is DC?

( ) ( )

2682

68 2 22

16 610

AB DCEF

DC

DC

DCDC

+=

+=

+=

= +=

OR

6 10 8

10´ You try: a) The sum of the measures of the five angles of a pentagon is the sum of the measures of the angles of the three triangles: OR

( )( )

2 180

5 2 180540

n= − °

= − °= °

b) n stands for the number of sides of the polygon.

11´ You try:

11) Each interior angle of a regular polygon has measure 120º. How many sides does the polygon have?

( 2) 180n− ⋅ is the sum of all of the interior

angles, so ( 2) 180nn

− ⋅ is the measure of one

of the interior angles:

( 2) 180 120nn

− ⋅ =

Or the sum of the measures of the interior angles is 120° for each angle times the number of angles.

( 2)180 120180 360 120

60 36060 36060 60

6

n nn n

nn

n

− =− =

=

=

=

D C

B A

E F

( )3 180 540° = °

28 122

20210

AB DCEF +=

+=

=

=

OR

8 12

10

Sum of measures of the interior ∠ ’s of a convex n-gon



12´ You try:

12) The apothem of a regular hexagon is

4 3 and the measure of each side is 8.

What is the area of the hexagon? A regular hexagon is made up of 6 congruent triangles. The base of each triangle is a side of the hexagon and the apothem is the height of each triangle: Area of 1 Triangle Area of the hexagon: OR Using the formula for a regular n-gon

A= 12aip

=124 3( ) 6i8( )

= 2 3 48( )= 96 3

The area of the hexagon is 296 3 units

13´ You try: 13) Find the apothem of a regular quadrilateral whose area is 12 square units.

A = 4iareaof 1triangle

12=4 12bh

!

"#

$

%&

12 = 4i122 3( ) a( )

12 = 2 2 3( ) a( )12 = 4 3 ia

3i44 3

=4 3 ia4 3

33= a

33i 33= a

3 = a

( )( )6 area of 1 triangle

6 16 3

96 3

A =

=

=

OR, because this is a

€

45 − 45 − 90 triangle,

( )

121 2 323

a x=

=

=

x

a

x

A = x ⋅ x

12 = x2

12 = x

2 3 = x

AΔ =1

2bh

=1

28( ) 4 3( )

= 4 4 3( )= 16 3



14´ You try:

14a) The radius of a circle is 9 and its

circumference is 18π . The area of the

circle is: a)

A = πr2

= π 9( )2

= 81π

The area of the circle is

€

54π square units. b)

A = πr2

54π = π r( )2

54ππ

=πr2

π54 = r2

54 = r

9i6 = r

3 6 = r

The radius of the circle is 3 6 units.

15´ You try:

15) True or false?

a. All rectangles are parallelograms. b. All parallelograms are rectangles.

c. All squares are rectangles. d. All squares are rhombuses. a. True. b. False. Parallelograms can be rectangles, rhombuses, and/or squares. c. True. d. True. A rhombus is a quadrilateral with all sides congruent. A square is a special kind of rhombus.

16´ You try: 16) Find the measure of each exterior angle

of a regular heptagon.

360360 77

÷ = .

The measure of each exterior angle is 3607

° ,

or 3517° .

17´ You try:

17) A trapezoid has a base that measures 43 and another base that measures 59. Its area is 1,020. Find its height.

A= 12h b1 +b2( )

1020 = 12h 43+59( )

1020 = 12102( )h

1020 = 51h102051

= h

20 = h

The height is 20 units.

18´ You try:

18) Find the volume of a right circular cylinder whose radius is 4 in and height is 7 in.

Base is a circle. A = πr2

€

∴B = πr2

V = Bih= πr2( )h= π 4 in( )

27 in( )

= π 16 in2( ) 7 in( )=112π in3

The volume of the right circular cylinder is 112π in3 .



19´ You try: 19) For a regular triangular pyramid, the edge of the base measures 6 and the lateral edges measure 5. Find the lateral area. The lateral area is the sum of the areas of the three triangular lateral faces. That is, lateral area= 3i areaof 1triangular lateral face( )

( )

13213 62

bh

h

⎛ ⎞= ⎜ ⎟⎝ ⎠⎛ ⎞= ⎜ ⎟⎝ ⎠

The height of each lateral face cuts the base of the isosceles triangle in half. The height is also perpendicular to the base and, therefore, forms a right triangle, so we can use the Pythagorean Theorem to find the height:

2 2 2

2 2 2

2 2 2

2

2

3 55 325 91644

a b chhhhhh

+ =+ =

= −= −===

The lateral area The lateral area is 36 square units.

20´ You try:

20) Find the lateral area of a right circular

cylinder if the height is 8 and the base has a circumference of 27. The lateral side of a right circular cylinder is shaped like a rectangle.

A = bh= circumference ih= 27 8( )= 216

∴ The lateral area is 216 square units.

6

5 h

( )( )

( )( )

13213 6 42

3 3 436

bh⎛ ⎞= ⎜ ⎟⎝ ⎠⎛ ⎞= ⎜ ⎟⎝ ⎠

==

Documents

HUSD High School Geometry Semester 2 Study Guide · 2013-05-15 · HUSD High School Geometry Semester 2 Study Guide Page 1 of 19 MCC@WCCUSD (HUSD) 03/09/13 1 ... The diagram shows