Let's solve the problem step by step:

We are tasked with finding the length of $BC$ (denoted as $x$) in the right triangle $\triangle ABC$ with altitude $BD$ drawn to the hypotenuse $AC$.

Key Information:

• $AC = 22$
• $DC = 8$
• $AD = AC - DC = 22 - 8 = 14$

Using Similar Triangles:

The altitude $BD$ splits $\triangle ABC$ into two smaller right triangles ($\triangle ABD$ and $\triangle CBD$), which are similar to $\triangle ABC$. This means we can use the geometric mean relationships:

1. $BD^2 = AD \cdot DC$
2. $BC^2 = AC \cdot DC$

Solving for $BC$:

From the second geometric mean property: $BC^2 = AC \cdot DC$ Substitute the known values: $BC^2 = 22 \cdot 8 = 176$ Take the square root: $BC = \sqrt{176} = \sqrt{16 \cdot 11} = 4\sqrt{11}$

Final Answer:

$BC = 4\sqrt{11}$

Would you like further details or explanations? Let me know!

Follow-up Questions:

1. How is the geometric mean property derived for right triangles?
2. What are the other lengths (like $BD$) in this triangle?
3. Can you use trigonometry to verify the answer for $BC$?
4. How does the similarity of triangles help solve other problems?
5. What is the general formula for splitting a hypotenuse using altitude?

Tip:

Always check the similarity of triangles and their proportional relationships when working with altitude in a right triangle!