Proving a Quadrilateral is a Trapezium Using Coordinate Geometry

In geometry, a trapezium (also known as a trapezoid) is a quadrilateral with at least one pair of parallel sides. When dealing with quadrilaterals in a coordinate plane, the properties of parallel lines can be established using their slopes. This article will guide you through the steps to prove that a quadrilateral ABCD is a trapezium using coordinate geometry.

Steps to Prove ABCD is a Trapezium

1. Identify the Coordinates

Let the coordinates of the vertices A, B, C, and D be as follows:

- Ax1, y1 - Bx2, y2 - Cx3, y3 - Dx4, y4

2. Calculate the Slopes

The slope m of a line through two points x1, y1 and x2, y2 is given by:

m (y_2 - y_1)/(x_2 - x_1)

Calculate the slopes of the pairs of opposite sides:

Slope of AB: mAB (y_2 - y_1)/(x_2 - x_1) Slope of CD: mCD (y_4 - y_3)/(x_4 - x_3) Slope of BC: mBC (y_3 - y_2)/(x_3 - x_2) Slope of AD: mAD (y_4 - y_1)/(x_4 - x_1)

3. Check for Parallelism

To determine if ABCD is a trapezium, check if at least one pair of opposite sides has equal slopes:

If mAB mCD, then sides AB and CD are parallel. If mBC mAD, then sides BC and AD are parallel.

Conclusion

If either condition is satisfied, i.e., one pair of opposite sides is parallel, then ABCD is a trapezium.

Example

Let's say the coordinates are:

A(1, 2) B(4, 2) C(5, 3) D(2, 3)

Calculate Slopes

- Slope of AB: mAB (2 - 2)/(4 - 1) 0 - Slope of CD: mCD (3 - 3)/(5 - 2) 0 - Slope of BC: mBC (3 - 2)/(5 - 4) 1 - Slope of AD: mAD (3 - 2)/(2 - 1) 1

Check for Parallelism

- mAB mCD 0 parallel - mBC mAD 1 parallel

In this example, both pairs of opposite sides are parallel, confirming that ABCD is a trapezium.

Alternative Method

Another way to determine if a quadrilateral is a trapezium by using the slopes of the lines:

Let Ax1, y1, Bx2, y2, Cx3, y3, and Dx4, y4 be the coordinates of the four points A, B, C, and D:

Calculate the slopes for each side using the formula (y2 - y1)/(x2 - x1) We examine the following two cases: First case: Determine the ratios r (y2 - y1)/(x2 - x1) and r' (y4 - y3)/(x4 - x3) Second case: Determine the ratios q (y3 - y2)/(x3 - x2) and q' (y4 - y1)/(x4 - x1)

To confirm that ABCD is a trapezium, we need to find:

r r' or q q'

These conditions will indicate that the quadrilateral ABCD has a pair of parallel sides, hence is a trapezium.

Conclusion

In summary, identifying the slopes of the sides in a coordinate plane and checking for equal slopes is a practical and efficient method to prove that a quadrilateral is a trapezium. By following the steps outlined in this article, you can easily solve problems related to this geometric shape using coordinate geometry.