Congruence and Transformations Chapter 7, Lesson 1

Congruent = Same Size & Shape Determine if the two figures are congruent by using transformations. Explain you reasoning. STEP 2: Translate ΔA’B’C’ until all sides and angles match ΔXYZ. STEP 1: Reflect ΔABC over a vertical line. So, the two triangles are congruent because a reflection followed by a translation will map ΔABC to ΔXYZ.

Example 1 Determine if these two figures are congruent by using transformations. Explain your reasoning. STEP 2: Translate the red figure up to the green shape. STEP 1: Reflect the red figure over a vertical line. Even if the reflected figure is translated up and over, it will not match the green figure exactly.

Got it? 1 a. b. Congruent; A reflection followed by a translation maps figure A onto Figure B. Not congruent; No transformations will match the two figures exactly.

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

Example 2 STEP 2: Translate the new image up. 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? STEP 2: Translate the new image up. STEP 1: Start with the pre-image. Rotate the letter “d” 180 about point A. 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.

Got it? 2 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. A vertical rotation followed by a translation. Yes, images produce by a reflection and translation are congruent.

Congruence Chapter 7, Lesson 2

Measure the sides and angles of each triangle and compare them. Real-World Link 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. What would Lauren need to do to show the two triangles are congruent? Measure the sides and angles of each triangle and compare them. 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.

Corresponding Parts In the figure below, we can proof why the two figures are congruent. 𝚫𝐀𝐁𝐂 ≅𝚫𝐃𝐄𝐅 The parts of each triangle that match, or correspond, are called corresponding parts.

Example 1 Write congruence statements comparing the corresponding parts in the congruent triangles. Corresponding Angles: ∠𝐉≅∠𝐆, ∠𝐋≅∠𝐈 ∠𝐊≅∠𝐇 Corresponding Sides: JK≅GH 𝐊𝐋≅𝐇𝐈 𝐋𝐉≅𝐈𝐆 **Make sure to add the line for the corresponding sides.**

Got it? 1 Write congruence statements comparing the corresponding parts in the congruent quadrilaterals.

Example 2 Triangle ABC is congruent to XYZ. Write congruence statements comparing the corresponding parts. Then determine which transformations map ΔABC toΔXYZ. The transformations from ΔABC to ΔXYZ consist of a reflection over the y-axis followed by a translation of 2 units down.

Stop and Reflect… You can determine which points correspond by using the congruence statements. If AB ≅ MN, then point A corresponds with point M.

Got it? 2 Parallelogram WXYZ is congruent to parallelogram KLMN. Write congruence statements. Determine which transformation(s) map parallelogram WXYZ to parallelogram KLMN. If you reflect KLMN over the x-axis and then translate it to the right 5 units, it coincides with WXYZ.

Finding Missing Measures 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? 2 feet

Got it? 3 Miley is using a brace to support a tabletop. In the figure, ΔBCE ≅ ΔDFG. If m∠CEB = 50, what is the measure of ∠FGD? Since ∠CEB and ∠FGD are corresponding parts in congruent figures, they are congruent. So, ∠FGD measures 50.

Similar Triangles Inquiry Lab

Measure and record the lengths and angles. ΔLPQ ΔLMN LP = m∠ L = LM = LQ = m∠ P = LN = m∠ M = PQ = m∠ Q = MN = m∠ N = 18 cm 78  9 cm 78 21 cm 58  58 10.5 cm 25 cm 44  12.5 cm 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. *Start with LPQ sides, then angles…then go to LMN**

Similarity and Transformations Chapter 7, Lesson 3

Vocabulary Rating Scale Copy 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) Word Know it Well Have seen or heard it No Clue What it means Dilation Scale Factor Similar polygons an enlargement or reduction of a figure the ratio of two similar figures two figures that have the same shapes, but different size

𝐻𝐺 𝐸𝐷 = 8 4 or 2 𝐺𝐼 𝐷𝐹 = 6 3 or 2 𝐼𝐻 𝐹𝐸 = 10 5 or 2 Two figures are a similar if the second can be obtained from the first by a transformation and dilation. Determine if the two triangles are similar by using transformations. 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. 𝐻𝐺 𝐸𝐷 = 8 4 or 2 𝐺𝐼 𝐷𝐹 = 6 3 or 2 𝐼𝐻 𝐹𝐸 = 10 5 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.

Example 1- Determine if the two rectangles are similar using transformations. STEP 1: Rotate rectangle VWTU 90 clockwise about W so that it is orientated the same way as rectangle WXYZ. STEP 2: Write ratios comparing the lengths of each side. 𝑊𝑇 𝑋𝑌 = 5 7 𝑇𝑈 𝑌𝑍 = 3 4 𝑈𝑉 𝑍𝑊 = 5 7 𝑉𝑊 𝑊𝑋 = 3 4 The ratios are not equal, so the two rectangles are not similar.

Got it? 1 a. b.

Scale Factor 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.

Example 2 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? Size of webpage photo: 2 in x 2 = 4 3 in x 2 = 6 Size of print: 4 in x 1.5 = 6 6 in x 1.5 = 9 The printed photo is a 6 x 9. All three photos are similar.

Got it? 2 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? Size of printed photo: 24 cm x 1.5 = 36 cm 30 cm x 1.5 = 45 cm Size of second photo: 36 cm x 3 = 108 cm 45 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

“triangle ABC is similar to triangle XYZ.” ΔABC ~ΔXYZ is read as “triangle ABC is similar to triangle XYZ.”

Example 1 – Determine if the figures are similar. Ask: Are the angles congruent? YES Then ask: Are the sides proportional? 𝐻𝐽 𝑀𝑁 = 7 10 𝐾𝐿 𝑃𝑄 = 7 10 𝐽𝐾 𝑁𝑃 = 1 2 𝐿𝐻 𝑄𝑀 = 1 2 Since 𝟕 𝟏𝟎 and 𝟏 𝟐 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 𝟖 𝟏𝟐 ≠ 𝟔 𝟖

Scale Factor 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 3 2 .

Example 2 Quadrilateral WXYZ is similar to quadrilateral ABCD. 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.

Example 2 Quadrilateral WXYZ is similar to quadrilateral ABCD. Find the missing measure. METHOD 1: Find the scale factor between the two figures. scale factor = 𝒀𝒁 𝑪𝑫 = 𝟏𝟓 𝟏𝟎 = 𝟑 𝟐 YZ is 1.5 times larger than CD. So, m would be 1.5 times larger than 12. m = 12(1.5) m = 18

Example 2 Quadrilateral WXYZ is similar to quadrilateral ABCD. Find the missing measure. METHOD 2: Setup a proportion to find the missing measure. 𝑿𝒀 𝑩𝑪 = 𝒀𝒁 𝑪𝑫 𝒎 𝟏𝟐 = 𝟏𝟓 𝟏𝟎 m ∙ 10 = 12 ∙ 15 10m = 180 m = 18

Got it? 2 Find each missing measure. WZ 19.5 AB 16

Similar Triangles and Indirect Measure Chapter 7, Lesson 5

Angle-Angle (AA) Similarity

Proof of the AA Similiary Rule In the figure below, ∠𝑋≅∠𝑃 and ∠𝑌≅∠𝑄 If you extend the lines, you can see the two triangles are similar.

Example 1 Determine whether the triangles are similar. If so, write a similarity statement. 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

Got it? 1 Determine whether the triangles are similar. If so, write a similarity statement. 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

Indirect Measure 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.

The street light is 13 feet tall. Example 2 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. ℎ𝑦𝑑𝑟𝑎𝑛𝑡 𝑠ℎ𝑎𝑑𝑜𝑤 𝑠𝑡𝑟𝑒𝑒𝑡 𝑙𝑖𝑔ℎ𝑡 𝑠ℎ𝑎𝑑𝑜𝑤 5 26 = 2.5 ℎ ℎ𝑦𝑑𝑟𝑎𝑛𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 𝑠𝑡𝑟𝑒𝑒𝑡 𝑙𝑖𝑔ℎ𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 5h = 26(2.5) 5h = 65 h = 13 The street light is 13 feet tall.

Example 3 In the figure at the right, triangle DBA is similar to triangle ECA. Ramon wants to know the distance across the lake. 𝐴𝐵 𝐴𝐶 = 𝐵𝐷 𝐶𝐸 320 482 = 40 𝑑 320d = 482(40) 320d = 19,280 d = 60.25 The distance across the lake is 60.25 meters.

Got it? 2 & 3 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? The telephone pole is 12 meters.

Slope and Similar Triangles Chapter 7, Lesson 6

Similar Triangles and the Coordinate Plane

Example 1 𝑨𝑪 𝑩𝑬 = 𝑩𝑪 𝑫𝑬 𝟔 𝟒 = 𝟑 𝟐 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. 𝑨𝑪 𝑩𝑬 = 𝑩𝑪 𝑫𝑬 𝟔 𝟒 = 𝟑 𝟐 Slope = 2

Got it? 1 Write a proportion comparing the rise to the run for each of the similar slope triangles shown above. Then find the numeric value. 𝑴𝑶 𝑵𝑶 = 𝑷𝑹 𝑸𝑹 = 𝟏 𝟐

Similar Triangles and Slope

The pitch of a roof refers to the slope of the roof line 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 𝒎= 𝒚𝟐 −𝒚𝟏 𝒙𝟐 −𝒙𝟏 𝒎= 𝟖 −𝟔 𝟏𝟐 −𝟖 = 𝟏 𝟐 𝒎= 𝒚𝟐 −𝒚𝟏 𝒙𝟐 −𝒙𝟏 𝒎= 𝟐 −𝟑 𝟎 −𝟐 = −𝟏 −𝟐 = 𝟏 𝟐

Got it? 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. m = 𝟏 𝟑

Area and Perimeter of Similar Figures Chapter 7, Lesson 7

Perimeter and Area of Similar Figures

Scale factor in Perimeter and Area The perimeters are related by a scale factor of “k”. The areas are related by a scale factor of “k2”.

The new perimeter of the new rectangle is 28 inches. Example 1 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 7 6 . The perimeter of the original is 24. x = 24( 𝟕 𝟔 ) x = 28. The new perimeter of the new rectangle is 28 inches.

Got it? 1 The new perimeter of Δ PQR is 48 meters. 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.

The perimeter of the actual garden is 576 inches, or 48 feet. Example 2 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? STEP 1: Scale factor = 18 𝑓𝑒𝑒𝑡 24 𝑖𝑛𝑐ℎ𝑒𝑠 = 216 𝑖𝑛𝑐ℎ𝑒𝑠 24 𝑖𝑛𝑐ℎ𝑒𝑠 =9 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.

The perimeter of TUVW is 24 inches Got it? 2 Two quilting squares are shown. The scale factor is 3:2. What is the perimeter of square TUVW? STEP 1: Scale factor = 3 2 STEP 2: Find the perimeter of TUVW. p = 16(1.5) p = 24 inches The perimeter of TUVW is 24 inches

The back porch will have an area of 160 square feet. Example 3 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. “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.

The larger mural will have an area of 3,456 square inches. Got it? 3 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. “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.