Q. 112

Use slopes to show that the quadrilateral whose vertices are $(1, - 1), (4, 1), (2, 2)$ and $(5, 4)$ is a parallelogram.

Verified

Answer

The given quadrilateral is a parallelogram.

1Step 1. Given information

Four points that are vertices of a quadrilateral are given: $(1, - 1), (4, 1), (2, 2)$ and $(5, 4)$ .

2Step 2. Required to find

Show that the given quadrilateral is a parallelogram.

3Step 3. Finding slopes of the sides of given quadrilateral

Slope, $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Slope of line joining $(1, - 1)$ and $(4, 1)$ is: $m_{1} = \frac{1 - (- 1)}{4 - 1}$

$m_{1} = \frac{2}{3}$

Slope of line joining $(2, 2)$ and $(5, 4)$ is: $m_{2} = \frac{4 - 2}{5 - 2}$

$m_{2} = \frac{2}{3}$

Slope of line joining $(4, 1)$ and $(5, 4)$ is: $m_{3} = \frac{4 - 1}{5 - 4}$

$m_{3} = 3$

Slope of line joining $(1, - 1)$ and $(2, 2)$ is: $m_{4} = \frac{2 - (- 1)}{2 - 1}$

$m_{4} = 3$

4Step 4. Checking whether opposite sides are parallel lines

As $m_{1} = m_{2}$ and $m_{3} = m_{4}$ , the slopes of opposite sides of quadrilateral are equal. So, the lines are parallel.

Therefore, given quadrilateral is a parallelogram.