Question

finding the perimeter of an isosceles triangle of known area
the flag is an isosceles triangle.
formula is used to calculate that the area is $\frac{1}{2}(5)(5)sin(36^{circ})approx7.35$ square feet.
recalling that this triangle is isosceles, we can write the area equation to solve for $x$.

Explanation:

Step1: Recall area formula for triangle

The area formula for a triangle with two - side lengths \(a\) and \(b\) and included angle \(\theta\) is \(A=\frac{1}{2}ab\sin\theta\). In an isosceles triangle with sides \(x\) and \(5\) and included angle \(\theta\), the area \(A = \frac{1}{2}(x)(5)\sin\theta\). We know \(A\approx7.35\) and \(\theta = 36^{\circ}\).

Step2: Solve for \(x\)

\[

$$\begin{align*} 7.35&=\frac{1}{2}(x)(5)\sin(36^{\circ})\\ 7.35&=\frac{5x}{2}\times0.5878\\ 7.35& = 1.4695x\\ x&=\frac{7.35}{1.4695}\approx5.0 \end{align*}$$

\]

Step3: Calculate the perimeter \(P\)

The sides of the isosceles triangle are \(x = 5.0\), \(5\) and \(5\). The perimeter \(P=x + 5+5\). Substituting \(x = 5.0\), we get \(P=5.0 + 5+5=15.0\) ft.

Answer:

The perimeter of the isosceles triangle is \(15.0\) ft.