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# Area of Trapezium – AMC 8, 2014 – Problem 14

The simplest example of power mean inequality is the arithmetic mean – geometric mean inequality. Learn in this self-learning module for math olympiad

## Area of Trapezium

A quadrilateral with at least one pair of parallel sides is called a trapezium. Area of a trapezium is half of the product of sum of the parallel sides and height. Let’s find its area as the sum of the area of a rectangle and a triangle.

## Try the problem

Rectangle (ABCD) and right triangle (\triangle DCE) have the same area. They are joined to form a trapezoid, as shown. What is (DE)?

American Mathematical Contest 2014, AMC 8  Problem 14

Area of Trapezium

3 out of 10

Challenges and Thrills in Pre College Mathematics

Excursion Of Mathematics

## Use some hints

Area of the triangle (\triangle DCE)=(\frac{DC.CE}{2}),

And the area of the square = (AB.AD)

It is given that (AB=5)

Then clearly (AB=DC=5)

(\frac{5.CE}{2}=30) , ((30) is the area of the square)

This gives the value of (CE)

Final Step

Now applying Pythagorean theorem in (\triangle ACE)

(DE^{2}=CE^{2}+DC^{2})

Which gives (DE=13)

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