Question

State whether the two lines through (9,5) and (-1,1) and through (3,-5) and (8,-3) are parallel, perpendicular or neither.

Answer 1

Hint: Find the slope of the lines using the property that the slope of the line joining the points $A\left( {{x}_{1}},{{y}_{1}} \right)$ and $B\left( {{x}_{2}},{{y}_{2}} \right)$ is given by $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$. Use the fact that if the slopes of two lines are equal, then they are parallel to each other and if the product of the slopes of two lines is -1, then the lines are perpendicular. Hence determine whether the lines are parallel or perpendicular or neither.

Complete step-by-step answer:
Finding the slope of the line joining (9,5) and (-1,1):
We know that the slope of the line joining the points $A\left( {{x}_{1}},{{y}_{1}} \right)$ and $B\left( {{x}_{2}},{{y}_{2}} \right)$ is given by $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$.
Here ${{x}_{1}}=9,{{x}_{2}}=-1,{{y}_{1}}=5$ and ${{y}_{2}}=1$
Hence the slope of the line is $m=\dfrac{1-5}{-1-9}=\dfrac{-4}{-10}=\dfrac{2}{5}$
Finding the slope of the line joining (3,-5) and (8,-3):
We know that the slope of the line joining the points $A\left( {{x}_{1}},{{y}_{1}} \right)$ and $B\left( {{x}_{2}},{{y}_{2}} \right)$ is given by $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$.
Here ${{x}_{1}}=3,{{x}_{2}}=8,{{y}_{1}}=-5$ and ${{y}_{2}}=-3$
Hence the slope of the line is $m=\dfrac{-3-\left( -5 \right)}{8-3}=\dfrac{2}{5}$
Now since the slope of the line joining (9,5) and (-1,1) is equal to the slope of the line joining (3,-5) and (8,-3), the lines are parallel to each other.
Note: [i] Viewing graphically:

As is evident from the graph $AB\parallel CD$
[ii] Alternative solution:
Let the equation of AB be y=mx+c
Since the line passes through (9,5), we have
$9m+c=5$
Also, since the line passes through (-1,1), we have
$-m+c=1$
Hence, we have
$9m+m=5-1\Rightarrow m=\dfrac{4}{10}=\dfrac{2}{5}$
Hence the slope of AB is $\dfrac{2}{5}$
Let the equation of CD be y = mx+c
Since the line passes through (3,-5), we have
$3m+c=-5$
Also, since the line passes through (8,-3), we have
$8m+c=-3$
Hence, we have
$8m-3m=-3+5\Rightarrow m=\dfrac{2}{5}$
Hence the slope of CD is $\dfrac{2}{5}$
Hence the lines are parallel to each other.