Question

Given right triangle ABC with altitude B D drawn to hypotenuse AC. If BD = 6 and
DC = 9, what is the length of AD?

Answer 1

4 units

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Use the geometric mean theorem (also called the altitude-on-hypotenuse theorem).

The theorem states that in a right triangle, the altitude from the right angle to the hypotenuse creates two similar right triangles with the larger triangle, and the length of the altitude is the geometric mean of the lengths of the two segments it creates on the hypotenuse.

Thus, we have triangle ABD and triangle CBD that are similar to triangle ABC. The geometric mean theorem gives us the following relationship:

• $\[ BD^2 = AD \cdot DC \]$

With the given lengths:

• $\[ 6^2 = AD \cdot 9 \]$

Now we solve for AD:

• $\[ 36 = AD \cdot 9 \]$
• $\[ AD = \cfrac{36}{9} \]$
• $\[ AD = 4 \]$

Therefore, the length of segment AD is 4 units.