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Tell whether the following statement is true or false:

In trapezium ABCD, AB || DC and L is the midpoint of BC. Through L, a line PQ ‖ AD has been drawn which meets AB in point P and DC produced in Q. Then area of trapezium ABCD = area of parallelogram APQD.

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True

False

answer is A.

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Detailed Solution

Given a trapezium.
In trapezium ABCD, AB ∥ DC and L is the mid - point of BC. Through L, a line PQ ∥ AD has been drawn which meets AB is P and DC

Since, L is mid-point on BC, CL = BL.
Now, in triangle BLP and CLQ,

∠ L C Q = ∠ L B P

(using alternate interior angle property)
CL=BL (given)

∠ C L Q = ∠ B L P

(using vertically opposite angle property)
Hence, BLP and CLQ are congruent using ASA postulate.
Now, using CPCT,

P B = C Q

. …(1)
Now, area of parallelogram APQD is,

a r e a = b a s e × h e i g h t a r e a = A P × D M

Area of trapezium ABCD is,

= 12 ×

(sum of parallel side)

×

(distance between them)

= 12 × A B + D C × D M

= 12 × A P + P B + D C × D M

= 12 × A P + D Q × D M (using equation 1)

= A P × D M

Therefore, area of trapezium ABCD = area of parallelogram APQD
Hence, the given statement is true.
Therefore, option 1 is correct.

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