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Understanding Right Triangle Altitude Properties

Learn the properties of altitude of a right triangle with Theorem 7.5 and Geometric Mean Theorems 7.6 and 7.7. Practice identifying similar triangles, finding values of variables, and solving geometric problems.

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Understanding Right Triangle Altitude Properties

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  1. Chapter 7.3 Notes: Use Similar Right Triangles Goal: You will use properties of the altitude of a right triangle.

  2. Theorem 7.5: If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. Ex.1: Identify the similar triangles in the diagram.

  3. Identify the similar triangles. Then find the value of x. Ex.2: Ex.3:

  4. Geometric Mean: • Theorem 7.6 Geometric Mean (Altitude) Theorem: In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments.

  5. Ex.4: Find the value of the variable.

  6. Theorem 7.7 Geometric Mean (Leg) Theorem: In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.

  7. Ex.5: Find the value of y. Write your answer in simplest radical form. Ex.6: Solve for x and y. x y 4 5

  8. Ex.7: Solve for x and y. 4 x y 12 Ex.8: Solve for x, y, and z. x y 12 16 z

  9. Ex.9: The diagram below shows a cross-section of a swimming pool. What is the maximum depth of the pool?

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