Question

In the following, three vertices of a rectangle are given. Plot these points on a graph paper and find the co-ordinates of the fourth vertex:

A(–1, 4), B(4, 4), C(4, –1)

Answer 1

1. Identify the given vertices and the structure of the rectangle

A rectangle is a quadrilateral where opposite sides are equal and parallel, and all internal angles are 90°. We are given three vertices of a rectangle ABCD:

A(–1, 4)

B(4, 4)

C(4, –1)

2. Determine the orientation of the known sides

Side AB: Points A(–1, 4) and B(4, 4) have the same y-coordinate (y = 4). This means ABis a horizontal line segment with a length of:

Length of AB

= |4 – (–1)| 

= 5 units

Side BC: Points B(4, 4) and C(4, –1) have the same x-coordinate (x = 4). This means BC is a vertical line segment with a length of:

Length of BC

= |4 – (–1)| 

= 5 units

3. Calculate the coordinates of the fourth vertex D

To complete the rectangle ABCD:

The side AD must be vertical, starting from A(–1, 4). Thus, D must have the same x-coordinate as A, which is –1.

The side CD must be horizontal, starting from C(4, –1). Thus, D must have the same y-coordinate as C, which is –1.

Alternatively, using the property that diagonals of a rectangle bisect each other, for A + C = B + D:

D = A + C – B

x_D = (–1) + 4 – 4

= –1

y_D = 4 + (–1) – 4

= –1

So, the coordinates of the fourth vertex are D(–1, –1).

4. Plotting the points

Below is the visualization of the rectangle ABCD on a coordinate plane.