The document discusses similar triangles and how to use proportions to solve problems involving similar triangles. It provides examples of setting up proportions between corresponding sides of similar triangles to determine unknown side lengths. It also gives examples of applying similar triangle proportions to solve real-world problems involving shadows.

1. Similar Triangles G.G.44 Establish similarity of triangles using the following theorems: AA, SAS, SSS G.G. 45 Investigate, justify, and apply theorems about similar triangles

2. Similar Triangles Unlike congruent triangles, which have the same angle AND side measures, SIMILAR triangles have the same angle measures, but the sides of one triangle may be bigger than the other! Example: 4 10 12 2 6 5 50º 50º 100º 100º 30º 30º

3. How it works: The lengths of the corresponding sides are in a proportion! So, you’d set up equations to show the proportion between the sides. 4 10 12 2 6 5 = = * Notice how the first triangle is always the numerator, and the second triangle is always the denominator. 50º 50º 100º 100º 30º 30º

4. In order to set up the proportions, you need to know what sides are corresponding. Just like congruent triangles, the corresponding sides are two sides that are between the same angles. 50º 50º 100º 100º 30º 30º A B C D E F So, since side AB is between the angles of 100º and 50º, you need to identify the side in the other triangle that is between the same angles! We would say that AB is similar to DE, BC is similar to EF, and CA is similar to FD.

5. Identify the similar sides of the following: 40º 60º 80º 80º 40º 60º G W Y N R L

6. Now, when you’re setting up proportions for similar triangles, remember that if you put the first triangle in the numerator, it has to ALWAYS be in the numerator for your equations. G W Y N R L

7. How would we use this to solve problems? If Δ ABC is similar to Δ DEF, and AB=9, DE=3, and BC=12, what is the measure of EF? Draw a picture! Label what you don’t know as x. Figure out your pairs of similar sides! Set up your proportion. Cross multiply to solve. 9 12 A B C D E F 3 x 9x = 36 x = 4

8. Example 2: Triangle NTE is similar to triangle KLA. If TE=16, EN=24, and AK=3, what is the length of LA? 48 = 24x 2 = x N T E K L A 16 24 3 x

9. The length of the shortest side of a triangle is 12, and the length of the shortest side of a similar triangle is 4. If the longest side of a triangle is 15, what is the longest side of a similar triangle? Cross-multiply to solve! 12x = 60 x = 5

10. A person 5ft tall is standing near a tree 30ft high. If the length of the person’s shadow is 3ft, what is the length of the shadow of the tree? person tree 5x = 90 x = 18 ft

11. A certain tree casts a shadow 6m long. At the same time, a nearby boy standing 2m tall casts a shadow 4m long. Find the height of the tree. Try some on your own! Three sides of a triangle are 3, 4 and 5. Find the length of the SHORTEST side of a similar triangle whose LONGEST side has a length of 20. longest side shortest side