Q. 8 - Geometry

Question

 This exercise provides a geometric method of justifying the fact that you can use any two points on a line to determine the slope of the line. Horizontal and vertical segments have been drawn as shown. Supply the reason for each step.

Key steps of proof:

$\begin{array}{l}1.\angle B\cong \angle A\\ 2.\angle 1\cong \angle 2\\ 3.\Delta LBN~\Delta DAJ\\ 4.\frac{BN}{AJ}=\frac{LB}{DA},or\frac{BN}{LB}=\frac{AJ}{DA}\end{array}$

5. The slope of LN equals $\frac{BN}{LB}$ and the slope of DJ equals $\frac{AJ}{DA}$

6. Slope of LN equals to slope of DJ.

Step-by-Step Solution

Answer

The reason for each step is shown below.

Step-1 – Given

Given that: We can use any two points on a line to determine the slope of the line.

Also given that the key steps of proof are:

$\begin{array}{l}1.\angle B\cong \angle A\\ 2.\angle 1\cong \angle 2\\ 3.\Delta LBN~\Delta DAJ\\ 4.\frac{BN}{AJ}=\frac{LB}{DA},or\frac{BN}{LB}=\frac{AJ}{DA}\end{array}$

5. The slope of $\overline{LN}$equals$\frac{BN}{LB}$ , and the slope of$\overline{DJ}$ equals$\frac{AJ}{DA}$ .

6. Slope of$\overline{LN}$ equals to slope of$\overline{DJ}$ .

Step-2 – To determine

We have to show reason for each step.

Step-3 – Calculation

In the Figure-(1), $\overline{LN}$and$\overline{DJ}$ are the two lines.

Step 1:

$\angle B\cong \angle A$

In the given figure we have that$\angle B=\angle A={90}^{\circ }$ .

So, the reason is: Both of the angles are right angles.

Step 2:

$\angle 1\cong \angle 2$

From Figure (1),$\overline{LB}\parallel \overline{AD}$

It means, the corresponding angles are equal.

So, the line$\overline{LB}$ is parallel to the line $\overline{AD}$, Hence the corresponding angles are equal.

Step 3:

We have to show that$\Delta LBN~\Delta DAJ$

In the$\Delta LBN$ and $\Delta DAJ$we have:

$\begin{array}{l}\angle B\cong \angle A\\ \angle 1\cong \angle 2\end{array}$

It means, they are the similar triangles.

Step 4:

$\frac{BN}{AJ}=\frac{LB}{DA},or\frac{BN}{LB}=\frac{AJ}{DA}$

From figure (1)$\Delta LBN~\Delta DAJ$ are the similar triangles. It means, the sides of the similar triangles are equal.

So, the corresponding sides are proportional because$\text{ΔLBN}$ is similar to$\text{ΔDAJ}$.

Step 5:

We know if a line makes an angle ${\text{n}}^{\text{o}}$ with the horizontal x-axis, then$\left(\text{m}\right)={\text{tan\hspace{0.33em}n}}^{\text{o}}$ .

So, tangent of the angle is${\text{tan\hspace{0.33em}n}}^{\circ }=\frac{\text{perpendicular}}{\text{base}}$

Substituting the values, we get:

Slope of$\overline{LN}$ is:

$\overline{LN}=\frac{BN}{LB}$

And,

Slope of$\overline{DJ}$ is

$\overline{LN}=\frac{AJ}{DA}$

So, the slope of$\overline{DJ}$ is$\frac{BN}{LB}$ similarly the slope of $\overline{DJ}$is $\frac{AJ}{DA}$.

Step 6:

From the given figure we have:$\angle 1\cong \angle 2$

So,$\tan \angle 1\cong \tan \angle 2$

We know that the slope of a line is given by the tangent of the angle.

The slope of the line (m) is:$\left(m\right)=\tan n^o$

So, the slope of the$\overline{LN}$ is equal to the slope of the $\overline{DJ}$.