Question

the flag is an isosceles triangle. the bottom side of the flag measures 5 feet because the triangle is a right triangle. because the triangle is a right triangle, the trigonometric triangle formula is used to calculate that the area is $\frac{1}{2}(5)(5)sin(36^{circ}) = 7.35$ square feet. recalling that this triangle is isosceles, we can write the area equation to solve for $x$. $\frac{1}{2}(x)(5)sin(\text{ })$ for $x$ and adding this value to the length of the sides, the perimeter = $\text{ }$ (rounded to the tenth).

Explanation:

Step1: Recall area formula for triangle

The area formula for a triangle is $A=\frac{1}{2}ab\sin C$. Given the area $A = 7.35$ square - feet, $a = x$, $b = 5$ and $C=72^{\circ}$ (since the triangle is isosceles, the non - base angles are $36^{\circ}$ each, and the included angle between the sides of lengths $x$ and $5$ is $72^{\circ}$). So, $7.35=\frac{1}{2}(x)(5)\sin(72^{\circ})$.

Step2: Solve for $x$

First, we know that $\sin(72^{\circ})\approx0.9511$. Then the equation becomes $7.35=\frac{1}{2}(x)(5)(0.9511)$.
\[

$$\begin{align*} 7.35&=\frac{4.7555x}{2}\\ 4.7555x&=7.35\times2\\ 4.7555x& = 14.7\\ x&=\frac{14.7}{4.7555}\approx3.1 \end{align*}$$

\]

Step3: Calculate the perimeter

The perimeter $P$ of the isosceles triangle with side lengths $x$, $x$, and $5$ is $P = 2x+5$. Substituting $x\approx3.1$ into the perimeter formula, we get $P=2\times3.1 + 5=6.2+5 = 11.2$ feet.

Answer:

The value of $x$ is approximately $3.1$ feet and the perimeter is $11.2$ feet.