Question

The base of an isosceles triangle has a length of 36 and a vertex anglemeasures 660. What is the length of each leg? Round to the nearest onehundredth of a foot.

Answer 1

The base angles of an isosceles triangle are equal. Thus,

`\begin{gathered} \angle A\text{ + }\angle B\text{ + }\angle C=180^0\text{ (sum of interior angles in a triangle)} \\ \angle B=\angle C\text{ ( base angles of an isosceles triangle are equal)} \\ \angle A\text{ + }\angle B\text{ + }\angle B\text{ = 180} \\ 66\text{ + 2}\angle B=180 \\ 2\angle B\text{ = }180\text{ - 66} \\ 2\angle B=\text{ 114} \\ \angle B=(114)/(2) \\ \angle B=57^0 \\ \text{Thus, }\angle C=57^0 \end{gathered}`

To get the leg AB, we will use the Sine rule:

`\begin{gathered} \frac{a}{\sin \text{ A}}=\frac{c}{sin\text{ C}} \\ \\ \frac{36}{\sin\text{ 66}}=\frac{c}{\sin \text{ 57}} \\ \\ c\text{ }*\text{ sin 66 = 36}*\sin \text{ 57} \\ c=\frac{36\text{ }*\text{ sin 57}}{\sin \text{ 66}} \\ c=33.05 \end{gathered}`

Hence, the length of leg AB IS 33.05

To get the leg AC, we use the sine rule too:

`\begin{gathered} \frac{b}{\sin\text{ B}}=\frac{a}{\sin \text{ A}} \\ \\ \frac{b}{\sin\text{ 57}}=\frac{36}{\sin \text{ 66}} \\ \\ b*\text{ sin 66 = 36 }*\text{ sin 57} \\ \\ b=\frac{36*\sin \text{ 57}}{\sin \text{ 66}} \\ b=33.05 \end{gathered}`

Hence, the length of the leg of AC is 33.05