Question

Find whether the lines drawn through the two pairs of points are parallel or perpendicular $(3,3),(4,6)$ and $(4,1),(6,7)$.

Answer 1

Let $A(3,3),$ $B(4,6),$ $C(4,1)$ and $D(6,7)$.

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 $A(3,3)$ and $B(4,6)$ as shown below:

$m_1=\frac{y_2-y_1}{x_2-x_1}=\frac{6-3}{4-3}=\frac{3}{1}=3$

Now, we find the slope of the line $CD$ with points $C(4,1)$ and $D(6,7)$ as shown below:

$m_2=\frac{y_2-y_1}{x_2-x_1}=\frac{7-1}{6-4}=\frac{6}{2}=3$

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 $m_1=m_2=3$

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