Q. 26 - Geometry

Question

a. Find the slopes of $\overline{OD}$ and$\overline{NF}$ .

b. Why is $\Delta OCD\cong \Delta NEF$? 

c.  Why is $\angle DOC\cong \angle FNE$?

d.  Why is$\overline{OD}\parallel \overline{NF}$ ?

e. What do you think is true about the slopes of parallel lines?

Step-by-Step Solution

Answer

1. The slope of$\overline{OD}$ and$\overline{NF}$ is (1).
2. $\Delta OCD\cong \Delta NEF$by SAS rule.
3. $\overline{OD}$and $\overline{NF}$ have the same angle with the$x\text{-axis}$ .
4. The line$\overline{OD}$ is parallel to the line$\overline{NF}$ .
5. The slope of the parallel lines is the same.

Part a. Step-1 – Given

Given figure is:

Step-2 – To determine

We have to find the slopes of$\overline{OD}$ and$\overline{NF}$ .

Step-3 – Calculation

In the figure (1):

Given coordinates of$OD$ are (0, 0) and (3, 3).

We have to calculate the slope of the line:

$m=\frac{y_2-y_1}{x_2-x_1}$

Substituting the values in the formula of slope for$\overline{OD}$ .

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

Given that the coordinates of $\overline{NF}$are (5, 0) and (8, 3).

We will calculate the slope of$\overline{NF}$ .

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

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

Part b. Step-1 – Given

Given figure is:

Step-2 – To determine

We have to show that $\Delta OCD\cong \Delta NEF$.

Step-3 – Calculation

In $\Delta OCD$ and $\Delta NEF$ in the given figure:

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

So, $\Delta OCD\cong \Delta NEF$

Hence, $\Delta OCD\cong \Delta NEF$ by SAS rule.

Part c. Step-1 – Given

Given figure is:

Step-2 – To determine

We have to show that $\angle DOC\cong \angle FNE$.

Step-3 – Calculation

From the given figure:

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

So,$\tan \angle DOC=\tan \angle FNE$

Hence, $\angle DOC\cong \angle FNE.$

Part d. Step-1 – Given

Given figure is:

Step-2 – To determine

We have to find why $\overline{OD}\parallel \overline{NF}$.

Step-3 – Calculation

In the given figure:

We see that$\overline{OD}$ and $\overline{NF}$ have the same angle with the , and it is also evident from the above solution.

$\tan \angle DOC=\tan \angle FNE$

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

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

Part e. Step-1 – Given

Given figure is:

Step-2 – To determine

We have to find the slope of parallel lines.

Step-3 – Calculation

In the given figure,

The parallel lines have the same angle with the .

So, the slopes of the parallel lines are the same.