Chapter 8.5, Problem 22PPS

a.

To determine

Label the side lengths and angle measures of triangles.

Answer to Problem 22PPS

  

  

  

Explanation of Solution

Given information:

Three triangles − acute, obtuse, right angle triangle.

Calculation:

Here, we will consider an acute triangle:

  

Now, we will consider an obtuse triangle:

  

Now, we will consider a right angled triangle:

  

Hence, the results are shown in the above figures.

b.

To determine

Complete the table.

Answer to Problem 22PPS

  $\begin{array}{llll}Triangle\hfill & Ratios\hfill & \hfill & \hfill \\ ABC\hfill & \frac{\sin A}{BC}=0.181\hfill & \frac{\mathrm{sinB}}{CA}=0.181\hfill & \frac{\mathrm{sinC}}{AB}=0.181\hfill \\ MNP\hfill & \frac{\sin M}{NP}=0.155\hfill & \frac{\mathrm{sinN}}{PM}=0.155\hfill & \frac{\mathrm{sinP}}{MN}=0.155\hfill \\ XYZ\hfill & \frac{\sin X}{YZ}=0.2\hfill & \frac{\mathrm{sinY}}{ZX}=0.2\hfill & \frac{\mathrm{sinZ}}{XY}=0.2\hfill \end{array}$

Explanation of Solution

Given information:

  $\begin{array}{llll}Triangle\hfill & Ratios\hfill & \hfill & \hfill \\ ABC\hfill & \frac{\sin A}{BC}\hfill & \frac{\mathrm{sinB}}{CA}\hfill & \frac{\mathrm{sinC}}{AB}\hfill \\ MNP\hfill & \frac{\sin M}{NP}\hfill & \frac{\mathrm{sinN}}{PM}\hfill & \frac{\mathrm{sinP}}{MN}\hfill \\ XYZ\hfill & \frac{\sin X}{YZ}\hfill & \frac{\mathrm{sinY}}{ZX}\hfill & \frac{\mathrm{sinZ}}{XY}\hfill \end{array}$

Calculation:

Here, we have to complete the given table:

  $\begin{array}{llll}Triangle\hfill & Ratios\hfill & \hfill & \hfill \\ ABC\hfill & \frac{\sin A}{BC}\hfill & \frac{\mathrm{sinB}}{CA}\hfill & \frac{\mathrm{sinC}}{AB}\hfill \\ MNP\hfill & \frac{\sin M}{NP}\hfill & \frac{\mathrm{sinN}}{PM}\hfill & \frac{\mathrm{sinP}}{MN}\hfill \\ XYZ\hfill & \frac{\sin X}{YZ}\hfill & \frac{\mathrm{sinY}}{ZX}\hfill & \frac{\mathrm{sinZ}}{XY}\hfill \end{array}$

Hence, the required answer will be:

  $\begin{array}{llll}Triangle\hfill & Ratios\hfill & \hfill & \hfill \\ ABC\hfill & \frac{\sin A}{BC}=0.181\hfill & \frac{\mathrm{sinB}}{CA}=0.181\hfill & \frac{\mathrm{sinC}}{AB}=0.181\hfill \\ MNP\hfill & \frac{\sin M}{NP}=0.155\hfill & \frac{\mathrm{sinN}}{PM}=0.155\hfill & \frac{\mathrm{sinP}}{MN}=0.155\hfill \\ XYZ\hfill & \frac{\sin X}{YZ}=0.2\hfill & \frac{\mathrm{sinY}}{ZX}=0.2\hfill & \frac{\mathrm{sinZ}}{XY}=0.2\hfill \end{array}$

c.

To determine

Prepare a conjecture about the ratio of the sine of an angle to the length of the opposite angle for a given triangle.

Answer to Problem 22PPS

The ratio of the sine of an angle to the length of the leg opposite that angle for a given triangle is constant.

Explanation of Solution

Given information:

Three triangles − acute, obtuse, right angle triangle.

Calculation:

Here, we have to prepare a conjecture.

Hence, the ratio of the sine of an angle to the length of the leg opposite that angle for a given triangle is constant.