Question

find x. (round your answer to four decimal places.) notice the inscribed triangle on the left is a right triangle. for this inscribed right tria in order to solve for its base? for the right triangle formed by the outer perimeter, w these two right triangles? resources ebook learn by example

Explanation:

Step1: Find base of inner - right triangle

Let the base of the inner right - triangle (with angle $57^{\circ}$) be $a$. Using the tangent function $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, we have $\tan(57^{\circ})=\frac{109}{a}$, so $a = \frac{109}{\tan(57^{\circ})}$.

Step2: Find base of outer - right triangle

Let the base of the outer right - triangle (with angle $35^{\circ}$) be $b$. Using the tangent function $\tan(35^{\circ})=\frac{109}{a + x}$, so $b=a + x=\frac{109}{\tan(35^{\circ})}$.

Step3: Solve for $x$

Since $b=a + x$ and $a=\frac{109}{\tan(57^{\circ})}$, $b = \frac{109}{\tan(35^{\circ})}$, then $x=\frac{109}{\tan(35^{\circ})}-\frac{109}{\tan(57^{\circ})}$.
We know that $\tan(35^{\circ})\approx0.7002$, $\tan(57^{\circ})\approx1.5399$.
$x = 109\times(\frac{1}{0.7002}-\frac{1}{1.5399})$
$x = 109\times(1.4282 - 0.6493)$
$x = 109\times0.7789$
$x\approx84.9001$

Answer:

$84.9001$