Question

one panel of a gable roof is in the shape of an isosceles triangle, whose sides measure 22 feet, 22 feet, and 30 feet. what is the area of this roof panel? enter your answer, rounded to the nearest tenth, in the box. ft²

Explanation:

Step1: Identify the semi - perimeter

Let \(a = 22\), \(b = 22\), \(c = 30\). The semi - perimeter \(s=\frac{a + b + c}{2}=\frac{22+22 + 30}{2}=\frac{74}{2}=37\) feet.

Step2: Use Heron's formula

Heron's formula for the area of a triangle is \(A=\sqrt{s(s - a)(s - b)(s - c)}\). Substitute \(s = 37\), \(a = 22\), \(b = 22\), \(c = 30\) into the formula:
\[

$$\begin{align*} A&=\sqrt{37(37 - 22)(37 - 22)(37 - 30)}\\ &=\sqrt{37\times15\times15\times7}\\ &=\sqrt{37\times15^{2}\times7}\\ &= 15\sqrt{37\times7}\\ &=15\sqrt{259}\\ &\approx15\times16.093\\ &=241.4 \end{align*}$$

\]

Answer:

241.4