Question

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

Answer 1

 

We are given the inradius formula:

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

Step 1: Using the given inradius formula

Substituting the given values into the formula: \[ 4 = \frac{P + 10 - H}{2} \] Multiply both sides by 2: \[ P + 10 - H = 8 \] Simplifying: \[ P - H = -2 \] So: \[ H = P + 2 \]

Step 2: Apply the Pythagorean Theorem

Using the Pythagorean Theorem: \[ P^2 - 10^2 = H^2 \] Substituting \( H = P + 2 \) into the equation: \[ P^2 - 100 = (P + 2)^2 \] Expanding: \[ P^2 - 100 = P^2 + 4 + 4P \] Simplifying: \[ 4P = 96 \] Solving for \( P \): \[ P = 24 \]

Step 3: Calculate the Area of the Triangle

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

Final Answer:

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