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

Correct Answer: 120

Applying Pythagoras Theorem and Area Calculation

We are given the equation from the Pythagoras theorem:

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

Step 1: Solving for \(x\)

First, expand both sides of the equation:

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

Now, expand the squares:

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

Simplifying:

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

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

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

Now, solve for \(x\):

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

\[ 80 = 4x \]

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

Step 2: Calculating the Area of the Triangle

We are given that the base \(b = 10\) and the height \(h = 24\). The area of the triangle is calculated using the formula:

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

Substitute the values of base and height:

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

Conclusion

The area of the triangle is 120 square inches.