Question

find the lengths of r, s, t, and u shown in the figure below if s + t = 26. round your answers to the nearest tenth. note that the figure is not drawn to scale.

Explanation:

Step1: Use tangent in small right - triangle

In the small right - triangle with side of length 9, $\tan(23^{\circ})=\frac{s}{r}$, so $s = r\tan(23^{\circ})$.

Step2: Use tangent in large right - triangle

In the large right - triangle, $\tan(23^{\circ})=\frac{s + t}{r}$. Since $s + t=26$, we have $\tan(23^{\circ})=\frac{26}{r}$. Then $r=\frac{26}{\tan(23^{\circ})}\approx\frac{26}{0.4245}\approx61.2$.

Step3: Find the value of s

Since $s = r\tan(23^{\circ})$ and $r\approx61.2$, then $s=61.2\times0.4245\approx26.0$.

Step4: Find the value of t

Since $s + t = 26$ and $s\approx26.0$, then $t=26 - s=26 - 26.0 = 0$.

Step5: Use cosine in small right - triangle to find u

In the small right - triangle, $\cos(23^{\circ})=\frac{r}{u}$, so $u=\frac{r}{\cos(23^{\circ})}\approx\frac{61.2}{0.9205}\approx66.5$.

Answer:

$r\approx61.2$
$s\approx26.0$
$t = 0$
$u\approx66.5$