Question

A line passing through the points $(2,7)$ and $(3,6)$ is parallel to a line joining $(9,a)$ and $(11,3)$. Find a.

Answer 1

Let $A(2,7),$ $B(3,6),$ $C(9,a)$ and $D(11,3)$.

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 $A(2,7)$ and $B(3,6)$ as shown below:

$m_1=\frac{y_2-y_1}{x_2-x_1}=\frac{6-7}{3-2}=\frac{-1}{1}=-1$

Now, we find the slope of the line $CD$ with points $C(9,a)$ and $D(11,3)$ as shown below:

$m_2=\frac{y_2-y_1}{x_2-x_1}=\frac{3-a}{11-9}=\frac{3-a}{2}$

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

Here, it is given that the lines $AB$ and $CD$ are parallel, therefore,$m_1=m_2$ that is:

$-1=\frac{3-a}{2}\Rightarrow -2=3-a\Rightarrow a=3+2=5$

Hence, $a=5$.