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Advanced Geometry LT 5.3 - Identify similar right triangles formed by the altitude drawn to the hypotenuse and use those properties to solve problems Finding Geometric Means of Pairs of Numbers The geometric mean of two positive numbers is the positive square root of their product. Example 1: Find the geometric mean of each pair of numbers. a) 4 and 25 b) 9 and 20 c) 5 and 12 Exploring the Geometric Mean Theorems Use tracing paper to compare the angles of triangles 1, 2, and 3. (Trace one of the triangles to start) What do you notice about the angles? What does this tell you about the triangles? Why? Similarity in Right Triangles The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. Geometric Mean Theorems The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg Example 2: Write similarity statements comparing the three triangles in each diagram. a) b) Example 3: Find the value of each variable. a) b) c) Practice Find the geometric mean x of each pair of numbers. Give answers in simplest radical form. 1. 3 and 12 2. 8 and 13 3. 2 and 8 Write a similarity statement comparing the three triangles in each diagram. 4. 5. Find the value of all variables. Give answers in simplest radical form. 6. 7. 8. 9. Round to the nearest tenth. 11. 10.