Q. 113

Use slopes to show that the quadrilateral

whose vertices are $(- 1, 0), (2, 3), (1, - 2)$ and $(4, 1)$ is a rectangle.

Verified

Answer

The given quadrilateral is a rectangle.

1Step 1. Given information

A quadrilateral with vertices $(- 1, 0), (2, 3), (1, - 2)$ and $(4, 1)$

2Step 2. Required to find

Show that the quadrilateral is a rectangle.

For this, show that slopes of the lines making opposite sides of quadrilateral are equal. Adjacent sides are perpendicular.

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

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

Slope of line joining vertices $(- 1, 0)$ and $(1, - 2)$ is: $m_{1} = \frac{- 2 - 0}{1 - (- 1)}$

$m_{1} = - 1$

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

$m_{2} = 1$

Slope of line joining vertices $(4, 1)$ and $(2, 3)$ is: $m_{3} = \frac{3 - 1}{2 - 4}$

$m_{3} = - 1$

Slope of line joining vertices $(2, 3)$ and $(- 1, 0)$ is: $m_{4} = \frac{0 - 3}{- 1 - 2}$

$m_{4} = 1$

4Step 4. Showing opposite sides are parallel

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

5Step 5. Showing adjacent sides are perpendicular

As $m_{1} \times m_{2} = - 1, m_{2} \times m_{3} = - 1, m_{3} \times m_{4} = - 1, m_{4} \times m_{1} = - 1$ , the adjacent sides are perpendicular.