Chapter 11.1, Problem 41SR

To determine

To find: The length of hypotenuse and of longer side in a ${30}^0-{60}^0-{90}^0$ triangle.

Answer to Problem 41SR

The measure of longer side is $12\surd 3$ feet and length of hypotenuse is 12 feet.

Explanation of Solution

Given information:

A triangle is ${30}^0-{60}^0-{90}^0$ right triangle with shorter side 12 feet.

Formulae used:

Use the following formula, to be applied in right triangle only, to get longer side,

  $\tan x=\frac{\text{perpendicular}}{\text{base}}$

Here x is angle of elevation. And hypotenuse is calculated using following Pythagorean Theorem:

  ${(\text{Hypotenues})}^2={(\text{Base})}^2+{(\text{Length})}^2$

Calculation:

By above concept, if angle of elevation is 30 degree, then shorter side 12 is its perpendicular and required longer side (say y ) is its base, so using first formula,

  $\begin{array}{l}\tan {30}^0=\frac{12}{y}\\ \frac{1}{\sqrt{3}}=\frac{12}{y}\end{array}$

Cross multiply,

  $y=12\sqrt{3}$

On using Pythagorean formula,,

  $\begin{array}{c}$ {(\text{Hypotenues})}^2={(12\sqrt{3})}^2+{(12)}^2$=144\times 3+144\\ \text{Hypotenues}=\sqrt{576}\\ \text{Hypotenues}=24\end{array}$

By above calculation, the measure of longer side is $12\surd 3$ feet and length of hypotenuse is 24 feet.