3. Similar Right Triangles
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Draw an altitude from the right angle and use the Angle-Angle (AA) Similarity Theorem to conclude that the three triangles formed are similar. Then, write the corresponding proportions and try to relate them to a geometric mean.
See solution.
Let's begin by considering a right triangle ABC, and let's draw the altitude from the right angle to the hypotenuse. Notice that the legs of the triangle are also altitudes.
Next, we will separate the three triangles formed above.
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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 of the hypotenuse. |
Next, let's write the proportions corresponding to the other two triangle similarities.
| Triangle Similarity | Proportions |
|---|---|
| â–³ ABC ~ â–³ ACD | AB/AC = AC/AD = BC/CD |
| â–³ ABC ~ â–³ CBD | AB/CB = BC/BD = AC/CD |
From the table above, we see that AC is the geometric mean of AB and AD. Also, BC is the geometric mean of AB and BD. This leads us to a second relation between altitudes and geometric means of a right triangle.
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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. |