Question

find x..(round your answer to four decimal places.) enter a number. 109 57° 35° x notice the inscribed triangle on the left is a right triangle. for this inscribed right triangle, in order to solve for its base? for the right triangle formed by the outer perimeter, what is these two right triangles? resources ebook learn by example

Explanation:

Step1: Find the base of the left - hand right - triangle

Let the base of the left - hand right - triangle be $a$. We know that $\tan(57^{\circ})=\frac{109}{a}$, so $a = \frac{109}{\tan(57^{\circ})}$.
Since $\tan(57^{\circ})\approx1.5399$, then $a=\frac{109}{1.5399}\approx70.7708$.

Step2: Find the base of the large right - triangle

Let the base of the large right - triangle be $b$. We know that $\tan(35^{\circ})=\frac{109}{a + x}$, so $a + x=\frac{109}{\tan(35^{\circ})}$.
Since $\tan(35^{\circ})\approx0.7002$, then $a + x=\frac{109}{0.7002}\approx155.6698$.

Step3: Solve for $x$

We know $a\approx70.7708$ and $a + x\approx155.6698$. Then $x=(a + x)-a$.
$x\approx155.6698 - 70.7708=84.8990$.

Answer:

$84.8990$