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Geometric mean

Geometric mean. When an altitude is drawn from the vertex of a right triangle's 90 degree angle to its hypotenuse, it splits the triangle into 2 right triangles that exhibit a special relationship. Theorem 50-1.

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Geometric mean

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  1. Geometric mean When an altitude is drawn from the vertex of a right triangle's 90 degree angle to its hypotenuse, it splits the triangle into 2 right triangles that exhibit a special relationship.

  2. Theorem 50-1 • If the altitude is drawn to the hypotenuse of a right triangle, then the 2 triangles formed are similar to the original triangle

  3. CAUTION !!! • Theorem 50-1 is only true if the altitude of the right triangle has an endpoint on the hypotenuse, not on the triangle's legs

  4. Identifying similar right triangles • Find RS and RQ,- Tri PQR is similar to Tri PSQ is similar to Tri RSQ P 3 S 5 4 R Q

  5. Finding geometric mean • Sometimes , the means of a proportion are equal to one another. • This is a special kind of proportion that can be used to find the geometric mean of 2 numbers • The geometric mean for positive numbers a and b, is the positive number x , such that

  6. Another way to state geometric mean • The geometric mean of a and b is equal to the square root of the product of a and b, since • ab = x2

  7. Find the geometric mean • Find the geometric mean of 3 and 12 • Find the geometric mean of 4 and 16 • Find the geometric mean of 2 and 9 in simplified radical form. • Find the geometric mean of 5 and 11 to nearest tenth.

  8. Corollary 50-1-1 • If the altitude is drawn to the hypotenuse of a right triangle, then the length of the altitude is the geometric mean between the segments of the hypotenuse

  9. Corollary 50-1-2 • If the altitude is drawn to the hypotenuse of a right triangle, then the length of a leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is closer to that leg. • AZ = YZ • YZ XZ

  10. Find missing values a and ba= 3b=43 5 4 5 4 3 b a 5

  11. Find missing value for y • 3 = y • y 4/3 y 3 4/3

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