Chapter 14.3, Problem 44PPSE

Solution

a.

To calculate: The value of $GH$using Pythagorean Theorem if in triangle $DGH,\angle H$ is a right angle, $h=13\text{ and }g=5$ .

The resultant answer is $GH=12$ .

Given information:

In triangle $DGH,\angle H$ is a right angle, $h=13\text{ and }g=5$ .

Formula used: The Pythagorean Theorem says that $c^2=a^2+b^2$ in a right-angle triangle. Here $a,b,\text{and }c$ represents the opposite, adjacent and hypotenuse respectively.

Calculation:

First draw the diagram using the given information:

  

Now use the Pythagorean Theorem to find the value of GH :

  $\begin{array}{l}G{D}^2=G{H}^2+H{D}^2\\ G{H}^2=G{D}^2-H{D}^2\end{array}$

Now substitute the values of $GD=13\text{ and }HD=5$ ,

  $\begin{array}{l}G{H}^2={\left(13\right)}^2-{\left(5\right)}^2\\ G{H}^2=169-25\\ G{H}^2=144\\ GH=12\end{array}$

Therefore, the value of $GH\text{ is 12}$ .

b.

To state: The value of $GH$ using a trigonometric ratio.

The resultant answer is $GH=12$ .

Given information:

In triangle $DGH,\angle H$ is a right angle, $h=13\text{ and }g=5$ .

Explanation:

The diagram is:

  

Now use the cosine function to find $\angle D$ .

  $\cos D=\frac{\text{adjacent}}{\text{hypotenus}}$

Now substitute the values of adjacent and hypotenuse.

  $\begin{array}{c}\cos D=\frac{\text{5}}{\text{13}}\\ \angle D={\cos }^{-1}\left(\frac{5}{13}\right)\\ \approx {67.38}^{\circ }\end{array}$

Now use the sine trigonometric.

  $\sin D=\frac{\text{opposite}}{\text{hypotenus}}$

Now substitute the values of opposite and hypotenuse.

  $\begin{array}{c}\sin D=\frac{x}{\text{13}}\\ x=13\sin \left(67.38\right)\\ x\approx 12\end{array}$

Therefore, the length of $GH\text{is }12$ .