Dilations Fill in the rows for "Dilations" in the Unit Focus chart. Is it an isometry? _____ _______________ (or angle measure) is preserved, but _______________ (or length) is not preserved. Therefore, a dilation is a similarity transformation. How is slope affected in a dilation? ____________________ Enlargements (or expansions) & reductions (or contractions) change both the ___ and ___ coordinates by the same ___________ ____________. However, when neither the shape nor size is preserved, the transformation is called a ____________________ Horizontal stretches & shrinks change the ___ coordinate. Vertical stretches & shrinks change the ___ coordinate. For enlargements and stretches, the scale factor is _______________ than _____. For reductions and shrinks, the scale factor is greater than _____ and less than _____.

Mapping Notation Description Image of (6, 0) Image of (-12, 3) 1. (x, y) → (2x, 2y)

2.

(x, y) → (13

x, 13

y)

3. (x, y) → (x,

12

y)

4. (x, y) → (

12

x, 1.5y)

Helpful strategies to find the scale factor for Dilations: write their coordinates, or find the slope, or find

the side lengths (or their corresponding right triangles on the grid). Remember, newscale factorold

=

5. At what coordinates should vertex Z be placed to create a quadrilateral WXYZ that is similar to quadrilateral PQRS? __________ What is the scale factor? ______

6. Triangle XYZ is dilated to form triangle X’Y’Z’. Find Z’ __________ What is the scale factor? ________

Figure 1 is dilated to produce Figure 2. In each of the following, identify the scale factor (SF) and the center of dilation.

7. SF: Center: 8. SF: Center: 9. SF: Center: 10. SF: Center:

Textbook page 511 #35-44 odds

Examples:

36. Glide reflection where the translation is (x, y) (x + 11, y) and the reflection line is y = 0. 40. Reflection across the line x = -0.5. 44. Rotation, 180° about center (0, 2).

Matching Activity – Transformations Name: ___________________________ Date: ________________ Period: _____ Find the Graph, Description, Equation in mapping notation, and example Point from each of these transformations and group them together. You will have 10 sets of 4 cards. Record your answers below:

(On the graphs, the pre-image is solid and the image is dotted)

Graph Description Equation Point G1

G2

G3

G4

G5

G6

G7

G8

G9

G10

Similar Polygons Name: ___________________________ Geometry 7-1 and 7-2 Date: ________________ Period: _____

Small Hexagon Large Hexagon Small / Large CB = RQ = CD = RS = DE = ST = EF = TU = FA = PU =

1. Using the ruler on the TAKS Formula Chart, measure (in centimeters) the lengths of all sides of the smaller hexagon and the larger hexagon.

2. Calculate the ratios of the smaller to larger hexagon for all corresponding sides. How do the ratios compare for the different sides of the hexagon? AB = PQ =

3. Using a protractor or patty paper, compare all

corresponding angles. How do the angles compare?

Two polygons are similar if and only if the corresponding angles are congruent and the corresponding sides are proportional. Similar objects have the same shape but not necessarily the same size. If two objects have congruent angles but not proportional corresponding sides, are they similar?

If two objects have proportional corresponding sides but not congruent angles, are they similar?

You can use the definition of similar polygons to find missing measures in similar polygons. Example: SMAL ~ BIGE. Find x and y.

Solution: The quadrilaterals are similar, so you can use a proportion to find x.

The measure of the side labeled x is 28 ft. In similar polygons, corresponding angles are congruent, so ∠M ≅ ∠I. The measure of the angle labeled y is therefore 83°. Notice that you could also use ratios within each quadrilateral to solve for x in this example: 18

2124 =x .

Check that you get the same result.

Class Exercises (taken from the Discovering Geometry textbook and workbook):

4. ∆PQS ∼ ∆QRS. What's wrong with this picture? 5. Are the pair of polygons similar? Explain why or why Explain. not.

In the figures below, state whether or not the figures are similar, and justify your answers. 6. ∆ABC and ∆ADE 7. JKON and JKLM 8. ABCD and AEFG

9. SPIDER ∼ HNYCMB. Find NY, YC, and MB. 10. Are these polygons similar? Explain why or why not.

11. ∆ACE ∼ ∆IKS. Find x and y. 12. ∆RAM ∼ ∆XAE. Find z. 14. ∆ABC ∼ ∆DBA. Find m and n.

13. THINK ∼ LARGE. Find AL, RA, RG, and KN. 3. Are these polygons similar? Explain why or why not.

Name: _____________________ Period: __________

Similar Triangles Anticipation Guide

Directions: Determine if you Agree or Disagree with each statement. Before

A/D Statement After A/D

If two figures are similar then they are congruent.

If the ratios of the length of corresponding sides of two triangles are equal, then the

triangles are congruent.

If triangles are similar then they have the same shape.

If two triangles are congruent then each pair of corresponding angles are congruent.

If two angles of one triangle are congruent to corresponding angles of another

triangle, then the triangles are similar.

Two polygons are similar if corresponding angles are congruent and corresponding

side lengths are proportional.

……………………………………………………………………………………………… Measure the angle to the nearest degree with a protractor.

Use the centimeter ruler from your TAKS formula chart to measure the segment below to the nearest one tenth of a centimeter.

Similar Triangles: AA, SAS, SSS Name: _____________________________ Geometry 7-3 Date: __________________ Period: _____ Warning: Diagrams are not drawn to scale! For each problem,

a. Determine whether the two triangles are similar. b. State the reason the two triangles are similar or not similar (AA, SSS, or SAS). c. Write the similarity statement in correct order. (Example: ∆ABC ∼ ∆EGF)

Some problems also ask for the value of x (which can only be found if you already know the ∆s are similar).

1. a) _____ b) __________ c) ∆ABC ∼ ∆__________

2. a) _____ b) __________ c) ∆SAM ∼ ∆__________

3. a) _____ b) __________ c) ∆APZ ∼ ∆__________

4. a) _____ b) __________ c) ∆ABC ∼ ∆__________

5. a) _____ b) __________ c) ∆JKL ∼ ∆__________

6. a) _____ b) __________ c) ∆ACB ∼ ∆__________

A S B C T U

R S A M J K

T A P Z R S

A D E B C F

M

J L

K N

7. a) _____ b) __________ c) ∆ABE ∼ ∆__________

8. a) _____ b) __________ c) ∆PHY ∼ ∆__________

Go back and check your work. All the pairs above are similar. Only #1 has two reasons why they are similar. Homework: Textbook Pg 386 #4-19. Pre-AP HW: Textbook Pg 385 # 2-16 evens, 17, 24-36 evens, 37, 44.

T P Y H A G

A 3 4 B E 5 C x = ________ D

9 2 4 x = ________ 3 6

A B 5 ________ = x C 6 5 D E 8

TOO TALL TO TELL Name_____________________________ Mirror Method GOAL: Use a mirror and similar triangles to determine the height of objects that cannot easily be

measured directly. MATERIALS: measuring tape or sticks, scientific calculator, mirror with cross hairs PROCEDURE: 1. Work in groups of 3: Recorder: ___________________________

Measurement taker: ___________________________

Observer: ___________________________

2. Draw a sketch that models your problem: 3. Determine what information is needed to find the height of the object.

Label your sketch with these variables. 4. Position the mirror between the observer and the object so that the observer can look in the mirror

and see the top of the object at the center of the crosshairs and the rest of the object appears aligned over one cross hair.

4. Gather data and fill in the first four columns of this chart - make sure to include units (you will calculate the height of the object in the next step):

Object to be

measured Distance from

object to mirror Eye height of

observer Distance from

observer Height of object

1.

2.

3.

4.

5. Write a proportion for computing the object’s height from your measurements. Define all variables used in the proportion. 6. Calculate the heights of each of the four objects and finish the chart. SHOW WORK HERE: Object 1

Object 2

Object 3

Object 4

7. Are there any errors that could have been made?

Application Problems using Similar Triangles 1. If a tree casts a 24-foot shadow at the same time that a yardstick casts a 2-foot shadow, find the height of the tree.

2. A bush is sighted on the other side of a canyon. Find the width of the canyon.

3. A 12-centimeter rod is held between a flashlight and a wall as shown. Find the length of the shadow on the wall if the rod is 45 cm from the wall and 15 cm from the light.

4. The cheerleaders at City High make their own megaphones by cutting off the small end of a cone made from heavy paper. If the small end of the megaphone is to have a radius of 2.5 cm, what should be the height of the cone that is cut off?

5. Find the width of the Brady River.

3 ft2 ft 24 ft

x ft

100 ft

10 ft

7.5 ft

x

shadow

60 cm

56 cm

2.5 cm

15 m

28 m 8 m

7 m 8 m

Brady River

6. The foot of a ladder is 1.2 m from a fence that is 1.8 m high. The ladder touches the fence and rests against a building that is 1.8 m behind the fence. Draw a diagram, and determine the height on the building reached by the top of the ladder.

7. Ramon places a mirror on the ground 45 ft from the base of a geyser. He walks backward until he can see the top of the geyser in the middle of the mirror. At that point, Ramon’s eyes are 6 ft above the ground and he is 7.5 ft from the mirror. Use similar triangles to find the height of the geyser.

8. Find the height of the giraffe in the diagram below.

9. On level ground, the base of a tree is 20 ft from the bottom of a 48-ft flagpole. The tree is shorter than the pole. At a certain time, their shadows end at the same point 60 ft from the base of the flagpole. How tall is the tree?

10. A tourist on the observation deck of a 400 ft building looks east, facing another building 320 ft high and two blocks from the first building. Her car is parked five blocks east of the second building. If no other buildings intervene, can she see her car?

x ft

45 ft 7.5 ft

6 ft

Tourist

Car

2 blocks 5 blocks

320 ft 400 ft

Lab Activity – Geometric Mean: Name:_________________ Right Triangle with Altitude to Hypotenuse Materials: Scissors Index card Straight Edge Procedure:

1) draw a diagonal on the index card, using a straight edge. 2) Cut on the diagonal, making two congruent right triangles

3) draw the altitude from the right angle to the hypotenuse. (segment from the vertex , perpendicular to the hypotenuse. Do on both triangles. )

e

b

e

c d

a

4) Label all segments as shown 5) Cut along the altitude on ONE of the right triangles. 6) Align the three triangles so that their corresponding sides are

similar. (This involves turning some of the triangles over)

7) Label all sides of triangles that have been turned over. 8) Complete the proportions. c = d c = b e = a d ? b = ? a ?

b

c d

a

a

b

e

c d

d b

c d

a d b

e

Geometry Name_________________________ Date__________ The Golden Ratio Activity

Rectanglesimilar tooriginalrectangle

Square

Golden Rectangle

A “golden rectangle” is a rectangle that can be divided into two parts: 1. A square 2. A rectangle that is similar to the original rectangle In any golden rectangle, the length and width of the rectangle compare in the “golden ratio” which is approximately 1.618 : 1. The golden ratio is interesting because it is considered to be very pleasing to the eye and shows up in architecture, art, and nature. Leonardo da Vinci (1492-1519) even wrote a book titled The Divine Proportion which is about the golden rectangle. American researcher Jay Hambridge has also found the Golden Ratio in the human body. Different proportions of the human body approximate the Golden Ratio. In this activity, we will measure to see if the Golden Ratio can be found in each of us. With your partner or group of 3, make each of the following measurements and fill in the table below. Measure to the nearest tenth of a centimeter.

Person’s name B N F K L H A E X Y

Now compute the following ratios for each person. Person’s name B:N F:K L:H A:E X:Y

Answer the following questions based on your data.

1. Which ratio best approximated the Golden Ratio for everyone? Why? 2. Which ratio was the worst approximation? Why? 3. With your partner or group, come up with another comparison of measurements of the body that you

think might approximate the golden ratio. Draw the comparison here (as on the front), make the measurements, compute the ratio and decide how close you were to the actual Golden Ratio.

19. a. Redraw all three triangles separately.

b. Make a chart. Small

leg Long leg

Hypo-tenuse

∆1

∆2

∆3

c. Set up a proportion. Solve for x. Use the Pythagorean Theorem to fill the chart completely.

35. a. Redraw all three triangles separately.

b. Make a chart. Small

leg Long leg

Hypo-tenuse

∆1

∆2

∆3

c. Set up a proportion. Solve for y. Solve for x and z using proportions or the Pythagorean Theorem.

90. a. Redraw all three triangles separately.

b. Make a chart. Small

leg Long leg

Hypo-tenuse

∆1

∆2

∆3

c. Use both the Pythagorean Theorem and proportions to solve for x and y.

92. At a golf course, Maria drove her ball 192 yd straight toward the cup. Her brother Gabriel drove his ball straight 240 yd, but not toward the cup. The diagram shows the results. Find x and y, their remaining distances from the cup.

a. Redraw all three triangles separately.

b. Make a chart. Small

leg Long leg

Hypo-tenuse

∆1

∆2

∆3

c. Set up a proportion and solve for x. Solve for y using either a proportion or the Pythagorean Theorem.

Answers are (#15) 9; (#17) 10; (#19) 12; (#35) x=12 5 , y=12, z=6 5 ; (#90) x=9, y=16; (#92) x=108 yd, y=180 yd.

9. AB || CD , AB = 4, AE = 3x + 4, CD = 8, and ED = x + 12. Find AE and DE. 10. , BE = 15, CF = 20, AE = 9, DF = 12 GFE GEF∠ ≅ ∠Determine which triangles in the figure are similar and why. (Write a similarity statement and give the reason or theorem.)

B

C

D E F

G

A

E

D

C

B

A

Hint: redraw the triangles separately and label them. Explain why each pair of triangles is similar and use the given information to find each missing measure. 11. 12.

S

A

X

T

E

9

6 3

4 2

x

R S

V T

U

x

y 12 9

3

20

13. In 2002, people in Laredo, Texas, erected the tallest flagpole in the United States. It can be seen from miles away. According to the information shown in the drawing, what is the approximate height of the flagpole? (The drawing is not drawn to scale.) Explain why these 2 triangles are similar.

14. To estimate the height of her school’s gym, Nicole sights the top of the gym wall in a mirror that she has placed on the ground. The mirror is 3.6 meters from the base of the gym wall. Nicole is standing 0.5 meter from the mirror, and her height is about 1.8 meters. What is the height of the gym wall?

Identify the similar triangles in each figure. Explain why they are similar (AA, SSS, SAS) and find the missing measures or values of “x” and “y”. 15. MN ll Δ________ ∼ Δ________

reason ________

AB = ________

BC = ________

BN = ________ A B

C

M N 12

N Q

S L

4 6

2

M P

K R

30

10

16. Δ________ ∼ Δ________

reason ________

16 16

8

B

x = ________

y = ________

17. Δ________ ∼ Δ________

reason ________

x = ________

y = ________

18. Δ________ ∼ Δ________

reason ________

x = ________

y = ________

Solve for x in the following diagrams.

19. 20. 21. 22.

Find the perimeter of the triangle. 23.

Trapezoid KMPR is similar to trapezoid LNQS.

Find the perimeter of KMPR.

A B C

D

E

x

C

A

D

E

y x 12

4 15

10

y

4 3

6

6

A B

C

D E y x

10 15

52

24

Part I: Slopes and Transformations – Name: ___________________________ Give the slope of the image line if the slope of its pre-image is given.

Transformation Describe how the slope of the original line is related to the slope of its image. 2

3/4 0 undefined - 5

a translation

a reflection over the x-axis

a reflection over the y-axis

a rotation of 90  o

a rotation of 180  o

a rotation of 270  o

a reduction

an enlargement

a reflection over the line y = x

Part II: Equations and Transformations – Write the equation of the image line after each of the transformations. Give your answer in slope-intercept form.

Transformation y = 1 y = –2 x = 4 x = –3 y = x y = x + 3 y = 2x – 6

a translation 2 down

a translation 4 units right and 3 units down

a translation 5 units left and 2 units up

a reflection over the x-axis (which is the line ______)

a reflection over the y-axis (which is the line ______)

a rotation of 90  with center at origin

o

a rotation of 180  with center at origin

o

a rotation of 270  with center at origin

o