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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.

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

((A)12\qquad(B)13\qquad(C)14\qquad(D)15\qquad(E)16)

Source

Competency

Difficulty

Suggested Book

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

First hint

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

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

Second Hint

It is given that (AB=5)

Then clearly (AB=DC=5)

Third Hint

(\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)

- https://www.cheenta.com/mensuration-area-of-triangle-amc-8-2015-problem-21/
- https://www.youtube.com/watch?v=eR5r0jfYX2Q

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