Question

Find whether the lines drawn through the two pairs of points are parallel or perpendicular $(5,2),(0,5)$ and $(0,0),(-5,3)$.

Answer 1

Let $A(5,2),$ $B(0,5),$ $C(0,0)$ and $D(-5,3)$.

We know that the slope of the line joining two points $(x_1,y_1)$ and $(x_2,y_2)$ is:

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

Let us first find the slope of the line $AB$ with points $A(5,2)$ and $B(0,5)$ as shown below:

$m_1=\frac{y_2-y_1}{x_2-x_1}=\frac{5-2}{0-5}=\frac{3}{-5}=-\frac{3}{5}$

Now, we find the slope of the line $CD$ with points $C(0,0)$ and $D(-5,3)$ as shown below:

$m_2=\frac{y_2-y_1}{x_2-x_1}=\frac{3-0}{-5-0}=\frac{3}{-5}=-\frac{3}{5}$

We also know that if the slope of two lines are equal that is $m_1=m_2$, then the lines are parallel and

if the slope of the two lines have the relation $m_1\times m_2=-1$, then the lines are perpendicular to each other.

Here, since slope of the lines $AB$ and $CD$ are equal that is $m_1=m_2=-\frac{3}{5}$

Hence, the line $AB$ is parallel to $CD$.