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  • Write true or false and justify your answer. ABCD is a parallelogram and X is the mid-point of AB. If ar (AXCD) = 24 cm^2, then ar (ABC) = 24 cm^2.

Write true or false and justify your answer.
ABCD is a parallelogram and X is the mid-point of AB. If ar (AXCD) = 24 cm2, then ar (ABC) = 24 cm2.

Answers (1)

Answer: [False]

Solution.
We have parallelogram ABCD and X is a mid point of AB.

Now, ar(ABCD) = ar(AXCD) + ar(\triangle XBC) …(1)
Draw XG perpendicular to CD
Area of  \parallel gm ABCD = (base) (corresponding altitude)

 = (AB) (GX) …(2)

Area of trapezium AXCD = \frac{1}{2} (Sum of parallel sides) (Distance between them)

= \frac{1}{2} (DC + AX) (GX)

Now, AX = \frac{1}{2}AB and  AB = CD   (Given)

Area of trapezium AXCD = \frac{1}{2} (AB + \frac{1}{2} AB) (GX)…(3)

Area of   \triangle ABC = \frac{1}{2}  (base) (height)

= \frac{1}{2} (AB) (GX)

= \frac{1}{2} (AB) (GX) …(4)

Now,
Area of trapezium AXCD = \frac{1}{2} (AB + \frac{1}{2}AB) (GX) = 24 cm^{2}
                                                                                                  (given)

\frac{3}{4}(AB) (GX) = 24 cm^{2}
\frac{1}{4}(AB)(GX)=8 cm^{2}
\frac{1}{2}(AB)(GX)=16 cm^{2}

From equation (4),
\frac{1}{2}(AB)(GX)=  Area of \triangle ABC=16cm^{2}

 Hence, the given statement is false.

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