Problem 4: Find the length of BD in right triangle ABC with given hypotenuse and segment lengths.

Final Answer:
$3\sqrt{7}$

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Explanation:
Okay, let’s solve this step by step.

We have a right triangle ABC, with the right angle at B. The altitude BD is drawn from B down to the hypotenuse AC. That means BD is perpendicular to AC, and it splits AC into two pieces: CD and DA.

We’re told:
- AC = 16 (the whole hypotenuse)
- CD = 7 (one part of the hypotenuse)

So, the other part, DA, must be:
16 – 7 = 9

Now here’s the cool trick: when you draw an altitude to the hypotenuse in a right triangle, it creates two smaller right triangles that are similar to each other and to the big triangle. And there’s a special rule:

> The altitude (BD) is the geometric mean of the two pieces of the hypotenuse (CD and DA).

That means:
BD² = CD × DA
BD² = 7 × 9
BD² = 63

Now take the square root:
BD = √63

But we need to simplify √63. Let’s break it down:
63 = 9 × 7
√63 = √(9×7) = √9 × √7 = 3√7

So, the length of BD is $3\sqrt{7}$.

That’s your answer!