## In trapezoid ABCD, points M and N are arbitrary points on bases AB and CD respectively. Find the area of the trapezoid, if area AABN=23 cm2

Question

In trapezoid ABCD, points M and N are arbitrary points on bases AB and CD respectively. Find the area of the trapezoid, if area AABN=23 cm2 and area ACDM=18 cm2.

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1. Remark

This is quite a nice little problem. It takes a minute or three to figure out the answer, and when you do, you will be certain that you have been tricked. It is a little like the egg of Columbus.

Solution

The Base of Triangle ABN is AB

The Base of Triangle CDM is CD

The height of both given triangles is h. That is the distance between the two parallel lines.

Area ABN = 1/2*AB * h = 23 cm^2

Area CDM = 1/2*CD * h = 18 cm^2

Now the Area of the trapezoid is

Area_Trapezoid = 1/2 * h (AB + CD)     Using the distributive property Remove the brackets.

Area_Trapezoid = 1/2*AB*h + 1/2*CD*h Did you notice something? Those terms are just the area of the triangles (written above.)

Area Trapezoid = 23 + 18 = 41 cm^2 <<<< Answer