Q. 7 - Geometry

Question

a. Find the slope of AB.

b. Find tan n°.

c. Consider the statement: If a line with positive slope makes an acute angle of n° with the x-axis, then the slope of the line is tan n°. Do you think this statement is true or false? Explain.

Step-by-Step Solution

Answer

1. The slope of is 1:5.
2. The value of is 1.5.
3. The statement is true.

Part a. Step-1 – Given

The given figure is:

Step-2 – To determine

We have to find the slope of $\overleftrightarrow{AB}$.

Step-3 – Calculation

Slope formula with two points$\left(x_1,y_1\right)\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}\left(x_2,y_2\right)$ is:

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

The given points are$A\left(1,0\right)$ and$B\left(3,3\right)$ .

$m=\frac{3-0}{\left(3\right)-\left(1\right)}=\frac{3}{2}=1.5$

So, the slope of$\overleftrightarrow{AB}$ is 1:5

Part b. Step-1 – Given

The given figure is:

Step-2 – To determine

We have to find $\tan n^{\circ }$.

Step-3 – Calculation

Slope formula with two points$\left(x_1,y_1\right)\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}\left(x_2,y_2\right)$ is:

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

Here$\theta$ = the angle that the lines makes with the positive direction of the$x-\text{axis}$ .

The given points$A\left(1,0\right)$ are and$B\left(3,3\right)$ .

$\begin{array}{l}\tan n^{\circ }=m=\frac{3-0}{\left(3\right)-\left(1\right)}=\frac{3}{2}=1.5\\ \tan n^{\circ }=1.5\end{array}$

So, the value of$\tan n^{\circ }$ is 1.5.

Part c. Step-1 – Given

The given figure is:

Step-2 – To determine

We have to determine whether the statement is true or false and explain.

Step-3 – Calculation

Slope formula with two points$\left(x_1,y_1\right)\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}\left(x_2,y_2\right)$ is:

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

The given points are$A\left(1,0\right)$ and$B\left(3,3\right)$ .

$m=\frac{3-0}{\left(3\right)-\left(1\right)}=\frac{3}{2}=1.5$

Also, 

$\tan n^{\circ }=\frac{\text{Perpendicular}}{\text{Base}}=\frac{3}{2}=1.5$

So, the given statement is true.