Question:medium

A circle of diameter 8 inches is inscribed in a triangle ABC where ∠ABC = 90°. If BC = 10 inches then the area of the triangle in square inches is

Updated On: Jan 15, 2026
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Correct Answer: 120

Solution and Explanation

Pythagorean Theorem and Area Calculation Applied

The Pythagorean theorem yields the equation:

\[ (x + 4)^2 + 10^2 = (x + 6)^2 \]

Step 1: Solve for \(x\)

Expand both sides of the equation:

\[ (x + 4)^2 + 100 = (x + 6)^2 \]

Expand the squared terms:

\[ (x^2 + 8x + 16) + 100 = x^2 + 12x + 36 \]

Simplify:

\[ x^2 + 8x + 116 = x^2 + 12x + 36 \]

Cancel \(x^2\) from both sides:

\[ 8x + 116 = 12x + 36 \]

Isolate \(x\):

\[ 116 - 36 = 12x - 8x \]

\[ 80 = 4x \]

\[ x = \frac{80}{4} = 20 \]

Step 2: Calculate Triangle Area

Given base \(b = 10\) and height \(h = 24\). The triangle's area is calculated as:

\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]

Substitute the values:

\[ \text{Area} = \frac{1}{2} \times 10 \times 24 = 120 \, \text{sq inches} \]

Conclusion

The triangle's area is 120 square inches.

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