8 1 Similarity in Right Triangles Warm UpPass

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8 -1 Similarity in Right Triangles Warm Up(Pass Back Papers & Add on a Separate Sheet) 1. Write a similarity statement comparing the two triangles. ∆ADB ~ ∆EDC Simplify. 2. 3. Solve each equation. 4. Holt Geometry 5. 2 x 2 = 50 ± 5

8 -1 Similarity in Right Triangles Holt Geometry

8 -1 Similarity in Right Triangles Holt Geometry

8 -1 Similarity in Right Triangles Example 1: Identifying Similar Right Triangles Write a similarity statement comparing the three triangles. Sketch the three right triangles with the angles of the triangles in corresponding positions. W Z By Theorem 8 -1 -1, ∆UVW ~ ∆UWZ ~ ∆WVZ. Holt Geometry

8 -1 Similarity in Right Triangles Check It Out! Example 1 Write a similarity statement comparing the three triangles. Sketch the three right triangles with the angles of the triangles in corresponding positions. By Theorem 8 -1 -1, ∆LJK ~ ∆JMK ~ ∆LMJ. Holt Geometry

8 -1 Similarity in Right Triangles Consider the proportion. In this case, the means of the proportion are the same number, and that number is the geometric mean of the extremes. The geometric mean of two positive numbers is the positive square root of their product. So the geometric mean of a and b is the positive number x such that , or x 2 = ab. Holt Geometry

8 -1 Similarity in Right Triangles Example 2 A: Finding Geometric Means Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 4 and 25 Let x be the geometric mean. x 2 = (4)(25) = 100 x = 10 Holt Geometry Def. of geometric mean Find the positive square root.

8 -1 Similarity in Right Triangles Holt Geometry

8 -1 Similarity in Right Triangles Example 3: Finding Side Lengths in Right Triangles Find x, y, and z. 62 = (9)(x) x=4 6 is the geometric mean of 9 and x. Divide both sides by 9. y 2 = (4)(13) = 52 y is the geometric mean of 4 and 13. Find the positive square root. z 2 = (9)(13) = 117 z is the geometric mean of 9 and 13. Find the positive square root. Holt Geometry

8 -1 Similarity in Right Triangles Find the geometric mean of each. 3. 3 and 27 4. 9 and 16 5. 4 and 5 Holt Geometry 6. 8 and 12

8 -1 Similarity in Right Triangles Lesson Quiz: Part I Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 1. 8 and 18 2. 6 and 15 Holt Geometry 12

8 -1 Similarity in Right Triangles Lesson Quiz: Part II For Items 3– 6, use ∆RST. 3. Write a similarity statement comparing the three triangles. ∆RST ~ ∆RPS ~ ∆SPT 4. If PS = 6 and PT = 9, find PR. 4 5. If TP = 24 and PR = 6, find RS. 6. Complete the equation (ST)2 = (TP + PR)(? ). TP Holt Geometry

8 -1 Similarity in Right Triangles Warm-Up Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form. 1. 8 and 18 2. 6 and 15 Holt Geometry 12