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7.4 Similarity in Right Triangles

7.4 Similarity in Right Triangles. In this lesson we will learn the relationship between different parts of a right triangle that has an altitude drawn in it. Geometric Mean. Before we look at right triangles we will examine something called the GEOMETRIC MEAN.

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7.4 Similarity in Right Triangles

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  1. 7.4 Similarity in Right Triangles In this lesson we will learn the relationship between different parts of a right triangle that has an altitude drawn in it

  2. Geometric Mean Before we look at right triangles we will examine something called the GEOMETRIC MEAN Ex. Find the geometric mean between 9 and 4. You could solve the proportion OR take the short cut x2=36 x=6 x=6 Geometric Mean: The number x such that , where a, b, and x are positive numbers If we solve we get x2=ab, so

  3. Geometric Mean Ex. Find the geometric mean between 10 and 15. You could solve the proportion OR take the short cut x2=150 Geometric Mean: The number x such that , where a, b, and x are positive numbers If we solve we get x2=ab, so

  4. Practice Problems • Put these two problems on your direction sheet • Find the geometric mean between 5 and 20 • Find the geometric mean between 12 and 15. Geometric Mean: The number x such that , where a, b, and x are positive numbers If we solve we get x2=ab, so

  5. 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.

  6. Geometric Mean with Altitude 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 6.75in 5.2 in 8.75in So, since 6.75 is the altitude, it is the geometric mean of 5.2 and 8.75

  7. Similarity in Right Triangles Ex. Find the values of x in the following right triangles. 9 7 x is the geometric mean of 9 and 7 x x 5 is the geometric mean of x and 3 5 3

  8. Practice Problems Put these three problems on your direction sheet. Find y in each picture. 4. 3. y 2 8 5. 9 19 y

  9. Geometric Mean Second Corollary to Theorem 7-3: The altitude to the hypotenuse separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the lengths of the hypotenuse and the length of the segment of the hypotenuse adjacent to the leg. 6 is the geometric mean of 3 and 12 3 is the part of the hypotenuse closest to side of 6. 12 is the whole hypotenuse 6 3 6

  10. Geometric Mean f is the geometric mean of 10 and 12 Example. 10 f 2 w is the geometric mean of 2 and 9 w 7 2

  11. Practice Problems Put these two problems on your direction sheet C C 7. Find w, j 8. Find w, j w 8 j j A A B B w D D 12 5 4

  12. THE END THE END

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