ABCD IS A TRAPEZIUM WITH PARELLEL SIDES AB=a AND CD=b E AND F ARE MIDPTS OF NON PARALLEL SIDES THE - Maths - Areas of Parallelograms and Triangles

Question

ABCD IS A TRAPEZIUM WITH PARELLEL SIDES AB=a AND CD=b E AND F
ARE MIDPTS OF NON PARALLEL SIDES THE RATIO OF ar(ABEF) =
ar(EFCD)

Harleen Birdi · Accepted Answer

Since E and F are mid point of AD and BC.

∴ EF will be parallel to both AB and CD and $EF = \frac{a + b}{2}$ .

(Here we have used the result, which can be seen by going through the following link: https://www.meritnation.com/ask-answer/question/pqrs-is-a-trapezium-with-pq-sr-e-and-f-are-the-midpoints-of/quadrilaterals/472355 )

Now, ABFE and CDEF both are trapezium.

ABFE is a trapezium with parallel sides AB and EF which are 'a' and $\frac{a + b}{2}$ respectively

$∴ Area (ABFE) = \frac{1}{2} (a + \frac{a + b}{2}) h'$ , where h' is height of trapezium ABFE

and similarly, $Area (EFCD) = \frac{1}{2} (b + \frac{a + b}{2}) h''$ , where h'' is height of trapezium EFCD

Since E is mid point of side AD and EF || AB

∴ EY must be the height of trapezium

∴ h' = h''

$Thus , \frac{Area (ABFE)}{Area (EFCD)} = \frac{\frac{1}{2} (a + \frac{a + b}{2}) h'}{\frac{1}{2} (b + \frac{a + b}{2}) h'} = \frac{3 a + b}{a + 3 b} = 3 a + b : a + 3 b$

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ABCD IS A TRAPEZIUM WITH PARELLEL SIDES AB=a AND CD=b E AND F ARE MIDPTS OF NON PARALLEL SIDES.THE RATIO OF ar(ABEF) = ar(EFCD)