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Notes Geometric Mean / Similarity in Right Triangles

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

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Notes Geometric Mean / Similarity in Right Triangles

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  1. Notes Geometric Mean / Similarity in Right Triangles I can use relationships in similar right triangles.

  2. 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

  3. 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 = 9x 8 = 3 8 8 can be divided by another Perfect Square, 4 = 3 4 x 2 = 3 x 2 2 = 6 2

  4. 3 x x 12 = Write a proportion. x2 = 36 Cross-Product Property Find the positive square root. 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. x2 = 36 x = 6 The geometric mean of 3 and 12 is 6.

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

  6. 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.

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

  8. Example Find the length of the altitude. 3 x = X x 6 18 = x2 3 6 √18 = x √9 ∙ √2 = x 3 √2 = x

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

  10. Example 3 Find the length of the leg. y 5 + 2 = y 2 7 y = y 2 14 = y2 √14 = y

  11. Similarity in Right Triangles Solve for x. 2 6 6 x = Write a proportion. 2x = 36 Cross-Product Property x = 18

  12. x y y 2 + x = Write a proportion. y 2 + 18 18 y = Substitute 18 for x. y2 = 360 Cross-Product Property. y = 360 Find the positive square root. y = 6 10 Write in the simplest radical form. Similarity in Right Triangles Solve for y.

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