Chapter 14.4, Problem 17PPSE

Solution

To calculate: The perimeter of the triangle if one of the congruent side of an isosceles triangle is 10 cm long and one of the congruent angle has a measure of ${54}^{\circ }$ .

The resultant answer is 32 cm.

Given information:

One of the congruent side of an isosceles triangle is 10 cm long and one of the congruent angle has a measure of ${54}^{\circ }$ .

Formula used: The perimeter of the isosceles triangle is $P=2a+b$ .

Calculation:

Draw an isosceles triangle using the given information:

  

The side ‘ a’ is congruent side of an isosceles triangle d ‘ b’ is remains side.

Now find angle A ,

  $\begin{array}{c}m\angle A={180}^{\circ }-{54}^{\circ }-{54}^{\circ }\\ ={72}^{\circ }\end{array}$

Now use law of sine’s to find b ,

  $\frac{\sin {54}^{\circ }}{10}=\frac{\sin {72}^{\circ }}{b}$

Isolate b ,

  $\begin{array}{c}b=10\cdot \frac{\sin {72}^{\circ }}{\sin {54}^{\circ }}\\ =11.8\end{array}$

The perimeter of isosceles triangle: $P=2a+b$

Substitute the values of $a\text{ and }b$ ,

  $\begin{array}{c}P=2\left(10\right)+11.8\\ =20+11.8\\ \approx 32\end{array}$

Therefore, the perimeter of the triangle is 32 cm.

Yes, drawing a diagram helped in solving the problem.

The length of the third side of the isosceles triangle is needed to find the triangle.

The third side of the triangle is calculated using the sine’s law.