Chapter 8.6, Problem 39PPS

To determine

To calculate:

The triangle and angel measures to the degree and side measures to the nearest tenth.

Answer to Problem 39PPS

The measurements of triangle are $m\angle C\approx {23}^{\circ },m\angle D\approx {67}^{\circ },m\angle E\approx {90}^{\circ }$ .

Explanation of Solution

Given information:

  

Concept used:

Law of cosines:

  $c^2=a^2+b^2-2ab\cos C$

Where,

  $a,b=$ sides

  $\cos C=$ angle

Calculation:

  

With the help of the above diagram it is easy to get the three sides of triangle and non-included angle. So, with the help of the Law of cosines get the missing angle.

First consider $E$ to be the angle.

Law of cosines:

  $c^2=a^2+b^2-2ab\cos C$

Where,

  $\begin{array}{l}a=20\\ b=48\\ c=52\\ \cos C=E\end{array}$

First substitute,

  ${\left(52\right)}^2={\left(20\right)}^2+{\left(48\right)}^2-2\left(20\right)\left(48\right)\cos \left(E\right)$

Now simplify,

  $2704=400+2304-1920\cos E$

Now solve for the $E$

  $0=\cos E$

Now, get the value of $E$ by using the inverse cosine ratio

  $E\approx 90$

Similarly, use the law of cosine to get the measure of $D$

First substitute,

  ${\left(48\right)}^2={\left(20\right)}^2+{\left(52\right)}^2-2\left(20\right)\left(52\right)\cos D$

Now simplify,

  $2304=400+2704-2080\cos D$

Now solve for the $D$

  $0.3846\approx \cos D$

Now, get the value of $D$ by using the inverse cosine ratio

  $D\approx 67$

The sum of the angles of a triangle is $180$ . So,

  $m\angle C=180-\left(90+67\right)=23$

Hence, the measurements of triangle are $m\angle C\approx {23}^{\circ },m\angle D\approx {67}^{\circ },m\angle E\approx {90}^{\circ }$