The Geometric Mean Theorem describes the relationship between the altitude to the hypotenuse of a right triangle and the two segments it creates on the hypotenuse. Specifically, the altitude is the geometric mean between these two segments. It also relates each leg of the right triangle to the adjacent segment of the hypotenuse and the entire hypotenuse.
In simpler terms, if you draw a line from the right angle of a right triangle to the hypotenuse at a right angle, that line's length squared is equal to the product of the two pieces of the hypotenuse it creates. Let's practice!
Match the terms with their definitions:
| Term | Definition |
|---|---|
| 1. Altitude | A. The average of two numbers found by multiplying them and then taking the square root. |
| 2. Hypotenuse | B. The side opposite the right angle in a right triangle. |
| 3. Right Triangle | C. A line segment from a vertex of a triangle perpendicular to the opposite side. |
| 4. Geometric Mean | D. A triangle containing one 90-degree angle. |
| 5. Leg | E. Either of the two sides that form the right angle in a right triangle. |
Complete the paragraph using the words: altitude, geometric mean, hypotenuse, segments, right triangle.
In a ______ ______, the ______ drawn to the ______ creates two ______ on the hypotenuse. The length of the altitude is the ______ ______ between the lengths of these two segments.
Explain in your own words how you can use the Geometric Mean Theorem to find the length of an altitude in a right triangle. Provide an example.
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