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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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Solution and Explanation

A circle with a diameter of 8 inches is inscribed within a right-angled triangle ABC, where the angle at B is 90 degrees.

The inradius formula provided is:

\[ r = \frac{P + B - H}{2} \] Where: - \( P \) represents the perimeter, - \( B \) represents the base, and - \( H \) represents the height of the triangle.

Step 1: Applying the Inradius Formula

Substitute the given values into the formula: \[ 4 = \frac{P + 10 - H}{2} \] Multiply both sides by 2: \[ P + 10 - H = 8 \] Rearrange the equation: \[ P - H = -2 \] Therefore: \[ H = P + 2 \]

Step 2: Utilizing the Pythagorean Theorem

Apply the Pythagorean Theorem: \[ P^2 - 10^2 = H^2 \] Substitute \( H = P + 2 \) into the equation: \[ P^2 - 100 = (P + 2)^2 \] Expand the equation: \[ P^2 - 100 = P^2 + 4 + 4P \] Simplify the equation: \[ 4P = 96 \] Solve for \( P \): \[ P = 24 \]

Step 3: Determining the Triangle's Area

The area \( A \) of the triangle is calculated as: \[ A = \frac{1}{2} \times B \times H \] Substitute the known values: \[ A = \frac{1}{2} \times 10 \times 24 = 120 \]

Final Answer:

The area of the triangle is \( \boxed{120} \) square inches.

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