Question

find the lengths of r, s, t, and u shown in the figure below if r + s = 38. round your answers to the nearest tenth. note that the figure is not drawn to scale. 23 s t u 13 42° r = s = t = u =

Explanation:

Step1: Find $r$ using tangent function

In the right - triangle with side 13 and angle $42^{\circ}$, $\tan42^{\circ}=\frac{13}{r}$. So, $r = \frac{13}{\tan42^{\circ}}\approx\frac{13}{0.9004}\approx14.4$.

Step2: Find $s$

Since $r + s=38$, then $s = 38 - r$. Substituting $r\approx14.4$, we get $s=38 - 14.4 = 23.6$.

Step3: Find $t$ using similar - triangles property

The two right - triangles are similar. The ratio of their sides is the same. Since the horizontal side of the upper triangle is 23 and the vertical side of the lower triangle is 13 and we know the relationship between the vertical sides of the two triangles ($r$ and $s$), we can also use the fact that the two triangles are similar. The ratio of the vertical sides of the two similar triangles is the same as the ratio of the horizontal sides. So, $t = 23$.

Step4: Find $u$ using Pythagorean theorem

In the upper right - triangle with vertical side $s = 23.6$ and horizontal side 23, by the Pythagorean theorem $u=\sqrt{23^{2}+23.6^{2}}=\sqrt{529 + 556.96}=\sqrt{1085.96}\approx32.9$.

Answer:

$r\approx14.4$
$s\approx23.6$
$t = 23$
$u\approx32.9$