NOTES GEOMETRIC MEAN / SIMILARITY IN RIGHT TRIANGLES I can use relationships in similar right triangles.

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NOTES GEOMETRIC MEAN / SIMILARITY IN RIGHT TRIANGLES I can use relationships in similar right triangles.

Simplifying Radicals  Perfect Squares – 1, 4, 9, 16, 25, 36, 49, 64, 81…  Find the largest Perfect Square that goes into the number evenly example: 72 The largest Perfect Square that goes into 72 is 36. = 36 x 2 = x 2 = 6 2

What if you picked 9 instead of 36?  If you pick a smaller Perfect Square you must reduce more than once. example: 72 9 is a Perfect Square that goes into 72 evenly, though not the largest = 9 x 8 = can be divided by another Perfect Square, 4 = 3 4 x 2 = 3 x 2 2 = 6 2

Geometric Mean Geometric Mean is the square root of the product of two values. If a, b, and x are positive numbers and, then x is called the geometric mean between a and b. Example : Find the geometric mean of 3 and 12. x 2 = 36 Cross-Product Property x = 6 Find the positive square root. The geometric mean of 3 and 12 is 6. =Write a proportion. 3x3x x 12 x 2 = 36

Similarity in Right Triangles Altitude – segment drawn from 90 degrees to the opposite side

Right Triangle Similarity Theorem - If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and each other.

Similarity in Right Triangles - Corollary 1 The length of the altitude of the right triangle is the geometric mean between the segments of the hypotenuse. seg1 alt seg2

Example 3 x = x 6 18 = x 2 √18 = x √9 ∙ √2 = x 3 √2 = x 36 X Find the length of the altitude.

Similarity in Right Triangles – Corollary 2 Each leg of the right triangle is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to the leg. SHAL leg hypotenuse

Example y = y 2 14 = y 2 7 y = y 2 √14 = y Find the length of the leg.

Solve for x. Similarity in Right Triangles = Write a proportion. 2x = 36 Cross-Product Property x = x6x

y 2 = 360Cross-Product Property. y = 360Find the positive square root. y = 6 10Write in the simplest radical form. Similarity in Right Triangles = Write a proportion. xyxy y 2 + x = Substitute 18 for x. 18 y Solve for y.