Question

Find the unknown measure. Round lengths to the nearest hundredth and angle measures to the nearest degree. Please help

Answer 1

Given:

AR=12 and AP=4.6,

Using Pythagorean theorem to find RP.

$RP^2=AR^2+AP^2$

Substitute AR=12 and AP=4.6, we get

$RP^2=12^2+4.6^2^{}$

$RP^2=165.16$

Taking square root on both sides, we get

$RP^{}=\sqrt[]{165.16}=12.8514590611$$RP=12.85$

Hence the unknown length is 12.85 units.

Given that angle A=90 degrees.

Using sine law.

$\frac{\text{RP}}{\sin A}=\frac{AR}{\sin P}=\frac{AP}{\sin R}$

Substitute known values, we get

$\frac{12.85}{\sin90^o}=\frac{12}{\sin P}=\frac{4.6}{\sin R}$

$\text{ Consider }\frac{12.85}{\sin90^o}=\frac{12}{\sin P}\text{.}$$\text{Use }\sin 90^o=1.$

$\frac{12.85}{1}=\frac{12}{\sin P}\text{.}$$\sin P=\frac{12}{12.85}$

$\sin P=0.93385214007$$\text{ Use sin }69.04348449=0.93385214007$

$\sin P=\sin 69.04348449$$P=69$

Hence we get

$m\angle P=69^o$

Using the triangle sum property, we get

$m\angle P+m\angle A+m\angle R=180^o$

Substitute known values, we get

$69^o+90^o+m\angle R=180^o$

$159^o+m\angle R=180^o$

$m\angle R=180^o-159^o$

$m\angle R=21^o$

Hence we get unknown angels

$m\angle R=21^o$

$m\angle P=69^o$

The unknown length is 12.85 units.