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Similarity in Right Triangles Theorem 7-3: The altitude to the hypotenuse of a right triangle divides the triangles into two triangles that are similar to the original triangle and to each other. Geometric Mean Review: How do we find the arithmetic mean of 3 and 27? a x Geometric Mean: The number x such that , where a, b, and x x b are positive numbers The geometric mean, in mathematics, is a type of mean or average, which indicates the central tendency or typical value of a set of numbers. Find the geometric mean of 3 and 27. Find the geometric mean of 4 and 18. Note: x ab Purpose of the Geometric Mean 1. The geometric mean can give a meaningful "average" to compare two companies. 2. The use of a geometric mean "normalizes" the ranges being averaged, so that no range dominates the weighting. 3. The geometric mean applies only to positive numbers.[2] 4. It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the human population or interest rates of a financial investment. Geometric Mean 6.75in 5.2 in 8.75in Corollary to Theorem 7-3: 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 Similarity in Right Triangles Find the values of x and y in the following right triangle. X Y Y 4 5 X 4 + 5 You Try One!!! Find the values of x and y in the following right triangle. Proof of Corollary to Theorem 7-3 C Given : Right triangle, ABC, with CD the altitude to the hypotenuse AD CD Prove : CD DB A Statements 1. Right triangle, ABC, with D B Reasons 1. CD the altitude to the hypotenuse 2. AD CD 3. CD DB 2. Altitude of rt. Δ to hypotenuse divides into 2 ~ Δs 3. Real World Connection As Marla arrives at the lake from the parking lot, she reads a sign that says she is 320m from the dock. How far is Marla from the information center? Kick it up a notch! Find the value of x in the following right triangle. 2x - 1 1 x Similarity in Right Triangles m1 m4 m7 m2 m6 m8 m3 m5 m9