Chapter 8.5, Problem 31STP

To determine

Toselect:

The measure of triangle’s smallest angle in degree.

Answer to Problem 31STP

The correct option is $B)22.62$ .

Explanation of Solution

Given information:

Ratio of sides of triangle $5:12:13$ .

Calculation:

The expression $a:b:c$ means that the ratio of the first two quantities is $a:b$ , the ratio of the

last two quantities are $b:c$ , and the ratio of the first and last quantities is $a:c$ .

Just as the ratio $5:12$ is equivalent to $5x:12x$ , the extended ratio can be written as $5x:12x:13x$ .

Notice that $\left\{5x:12x:13x\right\}$ is a Pythagorean triple. So, a triangle has sides in the ratio of $5:12:13$ is a right triangle.

Now, using:

Theorem of Angle-Side Relationships in a Triangle shows that the angle opposite the side of

Length $5x$ is the smallest angle.

Draw a figure to represent the given information.

  

From the definition of Sine:

If is a right triangle $\Delta ABC$ with acute $\angle A$ , then the sine of $\angle A$ is the ratio of the length of theleg opposite $\angle A$ to the length of the hypotenuse.

Thus, the definition of Sine shows that:

  $\begin{array}{l}\sin T=\frac{5x}{13x}\\ m\angle T={\sin }^{-1}\frac{5}{13}\\ m\angle T\approx 22.62\end{array}$

Hence, the correct option is $B)22.62$ .