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(5, 2), B(5, 5), C(1, 5)

Answer 1

The coordinates of the fourth vertex D are (1, 2).

To find the missing vertex of a rectangle when three vertices are provided, we look for the geometric relationships between the existing points:

1. Identify parallel and Perpendicular lines

In a rectangle, opposite sides are parallel and adjacent sides are perpendicular.

Side AB: The points A(5, 2) and B(5, 5) share the same x-coordinate (x = 5). This is a vertical line.

Side BC: The points B(5, 5) and C(1, 5) share the same y-coordinate (y = 5). This is a horizontal line.

2. Determine the missing coordinates

Since AB is vertical and BC is horizontal, the remaining sides must follow the same logic:

The side starting from C(1, 5) must be a vertical line to be parallel to AB. Therefore, vertex D must have the same x-coordinate as C, which is x = 1. 

The side starting from A(5, 2) must be a horizontal line to be parallel to BC. Therefore, vertex D must have the same y-coordinate as A, which is y = 2.

3. Verify the geometry

Combining these coordinates gives us D(1, 2).

The distance AB = |5 – 2| = 3 units.

The distance BC = |5 – 1| = 4 units.

The distance CD = |5 – 2| = 3 units (matches AB).

The distance DA = |5 – 1| = 4 units (matches BC).

The coordinates of the fourth vertex are D(1, 2).