Congruence and Transformations

Chapter 7, Lesson 1

Determine if the two figures are congruent by using transformations. Explain you reasoning.

Congruent = Same Size & Shape

STEP 1: Reflect ΔABC overa vertical line.

STEP 2: Translate ΔA’B’C’until all sides and angles match ΔXYZ.

So, the two triangles arecongruent because a reflection followed by a translation will map ΔABC to ΔXYZ.

Determine if these two figures are congruent by using transformations. Explain your reasoning.

Example 1

STEP 1: Reflect the red figure over a vertical line.

STEP 2: Translate the red figure up to the green shape.

Even if the reflected figure is translated up and over, it will not match the green figure exactly.

a.

b.

Got it? 1 Congruent; A

reflection followed by a translation maps

figure A onto Figure B.

Not congruent; No transformations will

match the two figures exactly.

If you have two congruent figures, analyze the orientation of the figures.

Determine the Transformations

Ms. Martinez created the logo shown. What transformations did she use if the letter “d” is the preimage and the letter “p” is the image? Are the two figures congruent?

Example 2

STEP 1: Start with the pre-image. Rotate the letter “d” 180 about point A.

STEP 2: Translate the newimage up.

Ms. Martinez used a rotation and

translation to create the logo. The letters

are congruent because rotation and

translation do not change the shape or

size.

What transformations could be used if the letter “W” is the preimage and the letter “M” is the image in the logo shown?

Are the two images congruent? Explain.

Got it? 2

A vertical rotation followed by a

translation.

Yes, images produce by a reflection and

translation are congruent.

CongruenceChapter 7, Lesson 2

Lauren is creating a quilt using the geometric pattern shown. She wants to make sure all of the triangles in the pattern are the same shape and size.

1. What would Lauren need to do to show the two triangles are congruent? Measure the sides and angles of each triangle

and compare them.

2. Suppose you cut out the two triangles and laid one on top of the other so the parts of the same measure were matched up. What is true about the triangles?

They are congruent.

Real-World Link

In the figure below, we can proof why the two figures are congruent.

The parts of each triangle that match, or correspond, are called CORRESPONDING PARTS.

Corresponding Parts

Write congruence statements comparing the corresponding parts in the congruent triangles.

Example 1

Corresponding Angles:

Corresponding Sides:

JKGH**Make sure to add the line for the corresponding sides.**

Write congruence statements comparing the corresponding parts in the congruent quadrilaterals.

Got it? 1

Triangle ABC is congruent to XYZ. Write congruence statements comparing the corresponding parts. Then determine which transformations map ΔABC toΔXYZ.

Example 2

The transformations from ΔABC to ΔXYZ consist of a reflection over the y-axis followed by a translation of 2 units down.

You can determine which points correspond by using the congruence

statements.

If AB MN,

then point A corresponds with point

M.

Stop and Reflect…

Parallelogram WXYZ is congruent to parallelogram KLMN. Write congruence statements. Determine which transformation(s) map parallelogram WXYZ to parallelogram KLMN.

Got it? 2

If you reflect KLMN over the x-axis and then translate it to the right 5 units, it coincides with WXYZ.

Miley is using a brace to support a tabletop. In the figure, ΔBCE ΔDFG if the length of CE is 2 feet, what is the length of FG?

Finding Missing Measures

2 feet

Miley is using a brace to support a tabletop. In the figure, ΔBCE ΔDFG. If mCEB = 50, what is the measure of FGD?

Got it? 3

Since CEB and FGD are corresponding parts in congruent figures, they are congruent.

So, FGD measures 50.

Similar TrianglesInquiry Lab

ΔLPQ ΔLMN

LP = m L = LM = m L =

LQ = m P = LN = m M =

PQ = m Q = MN = m N =

Measure and record the lengths and angles.

18 cm

21 cm

25 cm

78

58

44

9 cm

10.5 cm

12.5 cm

78

58

44

What do you notice about the measure of the corresponding angles of the triangles?

They are equal.

What do you notice about the ratios of the corresponding sides of the triangles?

They are equal.

Similarity and Transformations

Chapter 7, Lesson 3

WordKnow it Well

Have seen

or heard it

No Clue

What it means

Dilation

Scale Factor

Similar polygons

Vocabulary Rating ScaleCopy and complete the table. Place a check mark in the appropriate box next to the word. If you do not know the meaning, use your iPad to look up the word. (Use http://www.mathematicsdictionary.com)

an enlargement or reduction of a figure

the ratio of two similar figures

two figures that have the same shapes, but different size

Determine if the two triangles are similar by using transformations.

Two figures are a similar if the second can be obtained from the first by a transformation and dilation.

STEP 1: Translate ΔDEF down 2 units and 5 units to the right so D maps onto G.

STEP 2: Write ratios comparing the sides of each side. or 2 or 2 or 2

Since the ratios are equal, ΔHGI is a dilated image of ΔEDF. So, the two triangles are similar because of a translation and dilation.

STEP 1: Rotate rectangle VWTU 90 clockwise about W so that it is orientated the same way as rectangle WXYZ.

Example 1- Determine if the two rectangles are similar using transformations.

STEP 2: Write ratios comparing the lengths of each side.

The ratios are not

equal, so the two

rectangles are not

similar.

a.

b.

Got it? 1

Similar figures have the same shape, but have different sizes. They sizes of the two figures are related to the scale factor of the dilation.

Scale Factor

Ken enlarges the photo shown by a scale factor of 2 for his webpage. He then enlarges the webpage photo by a scale factor of 1.5 to print. If the original photo is 2 inches by 3 inches, what are the dimensions of the print? Are the enlarged photos similar to the original?

Example 2

Size of webpage photo:2 in x 2 = 43 in x 2 = 6 Size of print:

4 in x 1.5 = 66 in x 1.5 = 9

The printed photo is a 6 x 9. All three photos are similar.

An art show offers different size prints of the same painting. The original print measures 24 centimeters by 30 centimeters. A printer enlarges the original by a scale factor of 1.5, and then enlarges the second image by a scale factor of 3. What are the dimensions of the largest print? Are both prints similar to the original?

Got it? 2

Size of printed photo:24 cm x 1.5 = 36 cm30 cm x 1.5 = 45 cm

Size of second photo:

36 cm x 3 = 108 cm45 cm x 3 = 135 cm.

The largest size will be 108 x 135 cm. Yes, all three sizes are similar.

Properties of Similar Polygons

Chapter 7, Lesson 4

ΔABC ΔXYZ is read as “triangle ABC is similar to triangle

XYZ.”

Ask: Are the angles congruent? YES

Then ask: Are the sides proportional?

Example 1 – Determine if the figures are similar.

Since and are not equal, the rectangles are

not similar.

Got it? 1

Determine if the figures are similar.

No; the corresponding angles are not

congruent, and

A ratio of the lengths of two corresponding sides of two similar polygons.

Example: The two squares have a scale factor of 1.5 or .

Scale Factor

Quadrilateral WXYZ is similar to quadrilateral ABCD.

Example 2

a. Describe the transformation that map WXYZ onto ABCD.

Since the figures are similar and not congruent, a translation followed by a dilation would map WXYZ onto ABCD.

Quadrilateral WXYZ is similar to quadrilateral ABCD.

Example 2

b. Find the missing measure.

METHOD 1:Find the scale factor between the two figures.

scale factor =

YZ is 1.5 times larger than CD. So, m would be 1.5 times larger than 12.

m = 12(1.5)m = 18

Quadrilateral WXYZ is similar to quadrilateral ABCD.

Example 2

b. Find the missing measure.

METHOD 2:Setup a proportion to find the missing measure.

m ∙ 10 = 12 ∙ 1510m = 180

m = 18

Find each missing measure.

a. WZ19.5

b. AB16

Got it? 2

Similar Triangles and Indirect Measure

Chapter 7, Lesson 5

Angle-Angle (AA) Similarity

In the figure below, and

If you extend the lines, you can see the two triangles are similar.

Proof of the AA Similiary Rule

Determine whether the triangles are similar. If so, write a similarity statement.

Example 1

Angle A and E have the same measure. Since 180 – 62 – 48 = 70, G measures 70. Angle G and C have the same measure.

Two angles in ΔABC and ΔEFG are congruent.

ΔABC ΔEFG

Determine whether the triangles are similar. If so, write a similarity statement.

Got it? 1

Angle H and L have the same measure. Angle JKH and MKL are the same.

Two angles in ΔHJK and ΔLKM are congruent.

ΔHJK ΔLMK

Used to measure very large or very small items.

Since the two shapes are similar, then the angles are congruent.

We can use proportions to find the missing length.

Indirect Measure

A fire hydrant 2.5 feet high casts a 5 foot shadow. How tall is a street light that casts a 26-foot shadow at the same time? Let h represent the height of the street light.

Example 2

5h = 26(2.5)5h = 65h = 13

The street light is 13 feet tall.

In the figure at the right, triangle DBA is similar to triangle ECA. Ramon wants to know the distance across the lake.

Example 3

320d = 482(40)

320d = 19,280

d = 60.25

The distance across the lake is 60.25 meters.

At the same time a 2-meter street sign casts a 3-meter shadow, a nearby telephone pole casts a 12.3 meter shadow. How tall is the telephone pole?

Got it? 2 & 3

The telephone pole is 12 meters.

Slope and Similar Triangles

Chapter 7, Lesson 6

Similar Triangles and the Coordinate Plane

Write a proportion comparing the rise to the run for each of the similar slope triangles shown above. Then find the numeric value.

Example 1

Slope = 2

Write a proportion comparing the rise to the run for each of the similar slope triangles shown above. Then find the numeric value.

Got it? 1

Similar Triangles and Slope

The pitch of a roof refers to the slope of the roof line. Choose two points on the roof and find the pitch of the roof shown. Then verify that the pitch is the same by choosing a different set of points.

Example 2

The plans for a teeter-totter are shown at the right. Using points G and L, find the slope of the teeter-totter. Then verify that the slope is the same at a different location by choosing a different set of points.

Got it? 2

m =

Area and Perimeter of Similar Figures

Chapter 7, Lesson 7

Perimeter and Area of Similar Figures

The perimeters are related by a scale factor of “k”.

The areas are related by a scale factor of “k2”.

Scale factor in Perimeter and Area

Two rectangles are similar. One has a length of 6 inches and a perimeter of 24 inches. The other has a length of 7 inches. What is the perimeter of this rectangle?

The scale factor is . The perimeter of the original is 24.

x = 24()x = 28.

The new perimeter of the new rectangle is 28 inches.

Example 1

Triangle LMN is similar to triangle PQR. If the perimeter of ΔLMN is 64 meters, what is the perimeter of ΔPQR?

The new perimeter of Δ PQR is 48 meters.

Got it? 1

In a scale drawing, the perimeter of the garden is 64 inches. The actual length of AB is 18 feet. What is the perimeter of the actual garden?

Example 2

STEP 1: Scale factor = =

STEP 2: Find the perimeter of the actual garden. p = 64(9)p = 576

The perimeter of the actual garden is 576 inches, or 48 feet.

Two quilting squares are shown. The scale factor is 3:2. What is the perimeter of square TUVW?

Got it? 2

STEP 1: Scale factor =

STEP 2: Find the perimeter of TUVW. p = 16(1.5)

p = 24 inches

The perimeter of TUVW is 24 inches

The Eddingtons have a 5-foot by 8-foot porch on the front of their house. They are building a similar porch on the back with double the dimensions. Find the area of the back porch.

Example 3

“Double” means the scale factor is 2. The area of the front porch is (5)(8) or 40 square feet.

x = 40(22)x = 160

The back porch will have an area of 160

square feet.

Malia is painting a mural on her bedroom wall. The image she is reproducing is 4.8 inches by 7.2 inches. If the dimensions of the mural are 10 times the dimensions of the image, find the area of the mural in square inches.

Got it? 3

“Ten times” means the scale factor is 10. The area of the original image is 34.56 square inches.

x = 34.56(102)x = 3456

The larger mural will have an area of 3,456 square inches.