Chapter 8.6, Problem 42PPS

To determine

To calculate:

The $△XYZ$ for given measures.

Answer to Problem 42PPS

The measurements of triangle are $m\angle X\approx 74,m\angle Y\approx {23}^{\circ },m\angle Z\approx {83}^{\circ }$

Explanation of Solution

Given information:

Measurements of $△XYZ$ :

  $\begin{array}{l}XY=190\\ YZ=184\\ ZX=75\end{array}$

Concept used:

Law of cosines:

  $c^2=a^2+b^2-2ab\cos C$

Where,

  $a,b=$ sides

  $\cos C=$ angle

Calculation:

The given measurements of $△XYZ$ are $XY=190,YZ=184,ZX=75$ .

So, 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 $Y$ to be the angle.

Law of cosines:

  $c^2=a^2+b^2-2ab\cos C$

Where,

  $\begin{array}{l}a=190\\ b=184\\ c=75\\ \cos C=Y\end{array}$

First substitute,

  ${\left(75\right)}^2={\left(190\right)}^2+{\left(184\right)}^2-2\left(190\right)\left(184\right)\cos Y$

Now simplify,

  $5625=36100+33856-69920\cos Y$

Now solve for the $Y$

  $0.9201=\cos Y$

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

  $Y\approx 23$

Similarly, use the law of cosine to get the measure of angle $X$

First substitute,

  ${\left(184\right)}^2={\left(75\right)}^2+{\left(190\right)}^2-2\left(75\right)\left(190\right)\cos X$

Now simplify,

  $33856=5625+36100-28500\cos X$

Now solve for the $X$

  $0.27611\approx \cos X$

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

  $X\approx 74$

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

  $m\angle Z=180-\left(23+74\right)\approx 83$

Hence, the measurements of triangle are $m\angle X\approx 74,m\angle Y\approx {23}^{\circ },m\angle Z\approx {83}^{\circ }$