# Trapezoid ABCD is such that AB is parallel to CD and the diagonals intersect at point E...?

Triangle AEC's area is 20. Triangle CED is 50. What is the area of the entire trapezoid?

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• Anonymous
10 years ago

you mean Triangle AEB, because AEC is a straight line...

We can prove the 2 triangles are similar by the alternate interior angle thereom and the opposite angles thereom.

the ratio of area of AEB : area of CED = 2 : 5

Let the base of triangle CEB be B, and its height be H

the base and the height of AEB multiply TOGETHER, must be 2/5 of CED

so the shorter base is √ (2/5) or √2 / √5 or √10 / 5 of B or B√10 / 5

the height of AEB would be H√10 / 5

Area of any triangle is bh/2, so BH = 100, while 10BH / 25 = 40

Area of a trapezoid is (b1 + b2) h /2

or (B + B√10/5) (H + H√10/5) / 2

= (BH + BH√10/5 + BH√10 / 5 + 10BH / 25)/2

= (100 + 100√10 / 5 + 100√10 / 5 + 40)/2

= (140 + 40√10) / 2

= 70 + 20√10

the total area is 70 + 20√10

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• Anonymous
10 years ago

Based on this description, Triangle AEC isn't actually a triangle... Points A, E, & C are collinear... Could you check to make sure this is typed correctly?

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