Warm Up 1. Johnny’s mother had three children. The first child was named April. The second child was named May. What was the third child’s name? 2. A clerk at a butcher shop stands five feet ten inches tall and wears size 13 sneakers. What does he weigh? 3. Before Mt. Everest was discovered, what was the highest mountain in the world? 4. How much dirt is there in a hole that measures two feet by three feet by four feet? 5. What word in the English language is always spelled incorrectly?

Warm Up 1.) Dilate ∆𝑨𝑩𝑪 below by a scale of factor of 𝟐 𝟑 and list the new coordinates for ∆ 𝑨 ′ 𝑩 ′ 𝑪′ if ∆𝑨𝑩𝑪 has the following coordinates: A(3,6) B(6,6) C(9,9) 2.) Explain why ∆ 𝑨 ′ 𝑩 ′ 𝑪 ′ is a reduction of ∆𝑨𝑩𝑪. 3.) Give an example of a scale factor that would enlarge ∆𝑨𝑩𝑪. A B C A’ B’ C’ ∆ 𝐴 ′ 𝐵 ′ 𝐶′ is a reduction of ∆𝐴𝐵𝐶 because the scale factor (k) is less than 1. Correct Answer: Any number with a value greater than 1.

Similar Figures Mr. Riddle

I need your Help! My Problem: I’m enlarging a picture of my daughter Avery for my wife and need to make sure that every part of this picture will fit on the canvas. What’s the issue? Canvas’ come in different sizes and proportions and I need to figure out which one to buy.

What are Similar Figures? In Geometry, similar figures are figures that have the same shape and can be different in size. A similarity statement states that two figures are similar. Example: 𝐴𝐵𝐶𝐷~𝐸𝐹𝐺𝐻

How to Tell if Figures are Similar 𝐴𝐵𝐶𝐷~𝐸𝐹𝐺𝐻 If two figures are similar, two things MUST be true: 1.) All corresponding ANGLES are CONGRUENT. ∠𝐴≅∠𝐸; ∠𝐵≅∠𝐹; ∠𝐶≅∠𝐺; ∠𝐷≅∠𝐻 2.) All corresponding SIDES are PROPORTIONAL. 𝐴𝐵 𝐸𝐹 = 𝐵𝐶 𝐹𝐺 = 𝐶𝐷 𝐺𝐻 = 𝐷𝐴 𝐻𝐸 → 9 6 = 12 8 = 15 10 = 9 6 → 𝟑 𝟐 15 ft 10 ft 9 ft 6 ft 12 ft 8 ft 90° 60° 120°

Example 1 If ∆𝐹𝐺𝐻~∆𝐽𝐾𝐿, list all pairs of congruent angles, and write a proportion that relates all the corresponding sides. Use the similarity statement: ∆𝐹𝐺𝐻~∆𝐽𝐾𝐿 Congruent Angles: ∠𝐹≅∠𝐽; ∠𝐺≅∠𝐾; ∠𝐻≅∠𝐿 Proportion: 𝐹𝐺 𝐽𝐾 = 𝐺𝐻 𝐾𝐿 = 𝐻𝐹 𝐿𝐽

You Try! Congruent Angles: ∠𝐷≅∠𝐿; ∠𝐸≅∠𝑀; ∠𝐹≅∠𝑁 Proportion: If ∆𝐷𝐸𝐹~∆𝐿𝑀𝑁, list all the pairs of corresponding congruent angles and corresponding proportionate sides. Congruent Angles: ∠𝐷≅∠𝐿; ∠𝐸≅∠𝑀; ∠𝐹≅∠𝑁 Proportion: 𝐷𝐸 𝐿𝑀 = 𝐸𝐹 𝑀𝑁 = 𝐹𝐷 𝑁𝐿

Similarity Ratio The similarity ratio of similar figures is the ratio of the lengths of the corresponding sides of two similar polygons. The similarity ratio from ∆𝐴𝐵𝐶 to ∆𝑋𝑌𝑍 is 6:3 or 2:1. The similarity ratio from ∆𝑋𝑌𝑍 to ∆𝐴𝐵𝐶 is 3:6 or 1:2.

Help Me! Problem: I need to buy a canvas for my wife and I’m not sure which size to buy in order to fit the picture correctly. What information do you need from me? Should be at least 1.5 feet wide. Picture can’t be cropped. Dimensions of original picture are 4” by 5” Canvas sizes: 12’’ x 16” 20” x 30” 8” x 10” 11” x 17” 16” x 20” 5 inches 4 inches

Example 3: Similar Figures In the diagram: 𝑨𝑪𝑫𝑭~𝑽𝑾𝒀𝒁. Find x. Use the corresponding side lengths to make a proportion. 𝑪𝑫 𝑾𝒀 = 𝑫𝑭 𝒀𝒁 b. Find y. 𝑪𝑫 𝑾𝒀 = 𝑭𝑨 𝒁𝑽 𝟗 𝟔 = 𝟏𝟐 𝟑𝒚−𝟏 9 6 = 𝑥 10 𝒙=𝟏𝟓 27𝑦−9=72 27𝑦=81 𝒚=𝟑

You Try! 6𝑥−3 3 = 4 2 B. 3𝑦−2 5 = 4 2 12𝑥−6=12 6𝑦−4=20 12𝑥=18 6𝑦=24 Find the value of each variable if ∆𝑱𝑳𝑴~∆𝑸𝑺𝑻. Find x. Find y. 6𝑥−3 3 = 4 2 12𝑥−6=12 12𝑥=18 𝒙=𝟏.𝟓 B. 3𝑦−2 5 = 4 2 6𝑦−4=20 6𝑦=24 𝒚=𝟒

Perimeters of Similar Polygons If two polygons are similar, then their perimeters are proportional to the scale factor between them.

BREAK Take 7 minutes to use the restroom or stretch. End

∆𝐷𝐸𝐹~∆𝐴𝐵𝐶 Similar Triangles 3 in. 5 in. 10 in. 6 in. 6 in. 12 in. If two triangles are similar it means that: 1.) Their corresponding angles are congruent 2.) Their corresponding sides are proportional. 60° 45° 75° ∆𝐷𝐸𝐹~∆𝐴𝐵𝐶 3 in. 5 in. 10 in. 6 in. 6 in. 12 in.

Triangle Similarity

Example 4: Using AA~ Determine whether the triangles are similar. If so, write a similarity statement. Yes, since ∠𝐿≅∠𝑀 and ∠𝐾≅∠𝑄, the triangles are similar by AA similarity. ∆𝐽𝐾𝐿~∆𝑄𝑀𝑃 Yes, since ∠𝑅≅∠𝑊 and ∠𝑋≅∠𝑇, the triangles are similar by AA similarity. ∆𝑅𝑆𝑋~∆𝑊𝑆𝑇

You Try! Yes, they are similar by AA similarity. Determine whether the triangles are similar. If so, write a similarity statement. Yes, they are similar by AA similarity. ∆𝐿𝑃𝑄~∆𝐿𝐽𝐾 No, there are not two pairs of congruent angles.

SSS Similarity

SAS Similarity

Example 5: Using SSS and SAS Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. Triangles are similar by SAS or SSS. ∆𝑅𝑄𝑃~∆𝑅𝑇𝑆 Triangles are similar by SAS ∆𝐴𝐵𝐶~∆𝐴𝐹𝐸

You Try! Yes, ∆𝐽𝐾𝐿~∆𝑄𝑀𝑃 by the SSS Similarity. Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. Yes, ∆𝐽𝐾𝐿~∆𝑄𝑀𝑃 by the SSS Similarity. Yes, ∆𝑇𝑊𝑍~∆𝑌𝑊𝑋 by the SAS Similarity.

Assignment Aleks: HW#36