Chapter 13.2, Problem 26WE

(a)

To determine

To find: The slopes of $OD$ and $NF$ . $$

Answer to Problem 26WE

The slope of $\overline{OD}$ and $\overline{NF}$ is $\left(1\right)$ .

Explanation of Solution

Given information:

The given figure is,

  

Figure $\left(1\right)$

Calculation:

Consider the figure $\left(1\right)$

Given coordinates of $\overline{OD}$ are $\left(0,0\right)$ and $\left(3,3\right)$ .

Calculate slope of the line

  $\left(m\right)=\frac{y_2-y_1}{x_2-x_1}$

Substitute the values for the slope of $\overline{OD}$ .

  $\begin{array}{l}m=\frac{3-0}{3-0}\\ m=\frac{3}{3}\\ m=1\end{array}$

The coordinates of $\overline{NF}$ are $\left(5,0\right)$ and $\left(8,3\right)$ .

Calculate Slope of the $\overline{NF}$ .

  $\begin{array}{l}m=\frac{3-0}{8-5}\\ m=\frac{3}{3}\\ m=1\end{array}$

Therefore, the slope of $\overline{OD}$ and $\overline{NF}$ is $1$ .

(b)

To determine

To find: The $\Delta OCD\cong \Delta NEF$ .

Answer to Problem 26WE

The $\Delta OCD$ is congruent to $\Delta NEF$ by $\text{SAS}$ rule.

Explanation of Solution

Given information:

The given figure is,

  

Figure $\left(1\right)$

Calculation:

Consider the triangle $\Delta OCD$ and $\Delta NEF$ in the given figure $\left(1\right)$ .

  $\begin{array}{l}\overline{OC}=\overline{NE}\\ \overline{CD}=\overline{EF}\\ \angle C=\angle E\left({\text{90}}^{\text{o}}\right)\end{array}$

  $\Delta OCD\cong \Delta NEF$

Therefore, the $\Delta OCD$ is congruent to $\Delta NEF$ by $\text{SAS}$ rule.

(c)

To determine

To find: The $\angle DOC\cong \angle FNE$

Answer to Problem 26WE

The graph is increasing in the interval $x\le -0.5$ and decreasing in $x<-2$ and $x\ge -0.5$

  $$

Explanation of Solution

Given information:

The given figure is,

  

Figure $\left(1\right)$

Calculation:

Consider the figure $\left(1\right)$ .

Slope of $\overline{OD}$ is equal to the slope of $\overline{NF}$

So,

  $\tan \angle DOC=\tan \angle FNE$

Therefore, the angle $\angle DOC$ is equal to $\angle FNE$ .

(d)

To determine

To find: The $\overline{OD}\parallel \overline{NF}$ .

Answer to Problem 26WE

The line $\overline{OD}$ is parallel to the line $\overline{NF}$ .

Explanation of Solution

Given information:

The given figure is,

  

Figure $\left(1\right)$

Calculation:

Consider the given figure,

  $\overline{OD}$ and $\overline{NF}$ have the same angle with the $x\text{-axis}$ , and evident from the above solution.

  $\tan \angle DOC=\tan \angle FNE$

  $\overline{OD}\parallel \overline{NF}$

Therefore, line $\overline{OD}$ is parallel to the line $\overline{NF}$ .

(e)

To determine

To find: The slope of parallel lines.

Answer to Problem 26WE

The slope of the parallel lines are same..

Explanation of Solution

Given information:

The given figure is,

  

Figure $\left(1\right)$

Calculation:

Consider the figure,

The parallel lines have the same angle with the $x\text{-axis}$ .

Therefore, the slopes of the parallel lines are same.