Question

In ∆HIJ, i=99 inches and < H=9°. Find the length of h, to the nearest inch.

Answer 1

ANSWER

h = 16 in

EXPLANATION

We can solve this using the law of sines:

In this case, the relation is:

$\frac{i}{\sin I}=\frac{j}{\sin J}=\frac{h}{\sin H}$

WIth the first two ratios we have:

$\frac{99}{\sin I}=\frac{99}{\sin J}$

We can find that angles I and J are equal:

$\begin{gathered} \frac{\sin J}{\sin I}=\frac{99}{99} \\ \frac{\sin J}{\sin I}=1 \\ \sin J=\sin I \\ J=I \end{gathered}$

Therefore, they measures - because the interior angles of a triangle add up 180º- are:

$\begin{gathered} m\angle H+m\angle J+m\angle I=180º \\ 9º+2m\angle J=180º \\ m\angle J=\frac{180º-9º}{2} \\ m\angle J=m\angle I=85.5º \end{gathered}$

Now, using the law of sines, we can find h:

$\begin{gathered} \frac{i}{\sin I}=\frac{h}{\sin H} \\ \frac{99}{\sin85.5º}=\frac{h}{\sin 9º} \\ h=99\cdot\frac{\sin 9º}{\sin 85.5º} \\ h=15.535in \end{gathered}$

Rounded to the nearest inch, h = 16 in