Chapter 8.4, Problem 31WE

To determine

To explain:The reason any triangle having sides in the ratio $1:\sqrt{3}:2$ must be a $30°-60°-90°$ triangle.

Explanation of Solution

Given:

Thetriangle has sides in the ratio $1:\sqrt{3}:2$ .

Calculation:

The ratios are a Pythagorean triplet. Hence, one angle must be $90°$ .

Consider a right triangle with sides in the ratio $1:\sqrt{3}:2$

as shown below:

  

Take sine of angle ACB as follows:

  $\begin{array}{c}\sin \angle ACB=\frac{AB}{AC}\\ \sin \angle ACB=\frac{\sqrt{3}}{2}\\ \angle ACB=60°\end{array}$

Take sine of angle BAC as follows:

  $\begin{array}{c}\sin \angle BAC=\frac{BC}{AC}\\ \sin \angle BAC=\frac{1}{2}\\ \angle BAC=30°\end{array}$

Thus,any triangle having sides in the ratio as $1:\sqrt{3}:2$ must be a $30°-60°-90°$ triangle.