# 7.7.4. Pythagoras Theorem

In the following figure, is a trapezium where and . Find its area.

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### Example

In the following figure, is a trapezium where and . Find its area.

### Solution

Step 1: Find the length of .

Since is parallel to , we can find the length of by subtracting the length of from the length of .

Step 2: Find the length of .

Use Pythagoras' theorem, which states that the square of the length of a right-angled triangle's hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Substitute the known lengths in , and solve for the length of .

Step 3: Establish that quadrilateral is a rectangle.

is perpendicular to , so and are each equal to .

and are corresponding angles that are equal in measure, so is parallel to . Since is also parallel to , we know that quadrilateral is a parallelogram.

A parallelogram with a right angle must be a rectangle. Therefore, is a rectangle.

Step 4: Find the area of trapezium .

Add the area of and the area of rectangle .

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