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 an isosceles triangle, we can calculate that the base and height are known. solving for x and adding this value to the length of the other sides, the perimeter = (rounded to the nearest tenth)

Explanation:

Step1: Use tangent function

We know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here $\theta = 36^{\circ}$ and the adjacent - side to the $36^{\circ}$ angle is $x$ and the opposite - side is $5$. So $\tan36^{\circ}=\frac{5}{x}$.

Step2: Solve for $x$

We can rewrite the equation as $x=\frac{5}{\tan36^{\circ}}$. Since $\tan36^{\circ}\approx0.7265$, then $x=\frac{5}{0.7265}\approx6.9$.

Step3: Calculate the perimeter

The two equal sides of the isosceles triangle have length $\sqrt{x^{2}+5^{2}}$. Since $x\approx6.9$, the length of each of the equal sides is $\sqrt{(6.9)^{2}+5^{2}}=\sqrt{47.61 + 25}=\sqrt{72.61}\approx8.5$. The perimeter $P$ of the triangle is $P = 2\times8.5+5=17 + 5=22.0$.

Answer:

$22.0$