We are solving for the length of $BC$ in the right triangle $\triangle ABC$, where an altitude $\overline{BD}$ is drawn to the hypotenuse $\overline{AC}$. The problem provides:

• $AC = 8$
• $DC = 2$

Let $AD$ be the remaining segment of the hypotenuse $AC$. Since $AC = AD + DC$, we have: $AD = AC - DC = 8 - 2 = 6.$

Step 1: Use the Geometric Mean Theorem

The altitude $BD$ divides $\triangle ABC$ into two smaller triangles ($\triangle ABD$ and $\triangle CBD$) that are similar to each other and to $\triangle ABC$. According to the geometric mean theorem: $BD^2 = AD \cdot DC.$

Substitute the known values of $AD$ and $DC$: $BD^2 = 6 \cdot 2 = 12 \implies BD = \sqrt{12} = 2\sqrt{3}.$

Step 2: Apply the Pythagorean Theorem to $\triangle BCD$

In $\triangle BCD$, $BD$ is the altitude, $DC = 2$, and $BC$ is the hypotenuse. Using the Pythagorean theorem: $BC^2 = BD^2 + DC^2.$

Substitute $BD^2 = 12$ and $DC = 2$: $BC^2 = 12 + 2^2 = 12 + 4 = 16 \implies BC = \sqrt{16} = 4.$

Final Answer:

The length of $BC$ is: $\boxed{4}.$

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Do you want a detailed geometric diagram for this problem or further clarification?

Here are five related questions to deepen your understanding:

1. How does the geometric mean theorem relate to similar triangles in a right triangle?
2. What is the relationship between the segments of the hypotenuse when an altitude is drawn?
3. Can you calculate the length of $AB$ in this triangle using a similar method?
4. What happens to $BD$ if $DC$ is halved and $AC$ remains the same?
5. How would you solve this problem using trigonometric relationships instead of the geometric mean?

Tip: Always look for triangle similarity when an altitude is drawn to the hypotenuse—this simplifies many calculations!