Similarity in Right Triangles Students will be able to find segment lengths in right triangles, and to apply similarity relationships in right triangles to solve problems.

Unit F2 Geometric Mean Remember that in a proportion such as, a and b are called the extremes and r and q are called the means. The geometric mean of two numbers is the positive square root of their product. We use the following proportion to find the geometric mean: Notice that the means both have x. That is the geometric mean. How do you solve for x? →

Unit F3 Examples of Geometric Means Find the geometric mean between 4 and 9. → x 2 = 36 → → x = 6 Now, find the geometric mean between 2 and 8. → x 2 = 16 → → x = 4

Unit F4 Similar Right Triangles Theorem The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. C B D A ∆ABC ∼ ∆ACD ∼ ∆CBD

Unit F5 The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the two lengths of the segments of the hypotenuse. This means: The altitude is the geometric mean of the two segments of the hypotenuse and. Or we could say: CD 2 = AD ∙ BD C B D A Geometric Means Corollary

Unit F6 To estimate the height of a Douglas fir, Jan positions herself so that her lines of sight to the top and bottom of the tree form a 90º angle. Her eyes are about 1.6 m above the ground, and she is standing 7.8 m from the tree. What is the height of the tree to the nearest meter? Example of Corollary Set up the proportion: → 1.6x = 7.8 ∙ 7.8 x 1.6x = → x ≈ 38 So the tree is about 40 meters tall.