Question

ABCD is a parallelogram and X is the mid-point of AB. If ar(AXCD) = 24 cm² , then ar(ΔABC) = ________.

Answer 1

Given:
ABCD is a parallelogram
X is the mid-point of AB
ar(AXCD) = 24 cm²

We know, if a triangle and a parallelogram are on the same base and between the same parallels, then the area of the triangle is half the area of the parallelogram.
Thus, ar(ΔABC) = $\frac{1}{2}$ar(ABCD) ...(1)

Since, X is the mid-point of AB
Therefore, ar(ΔXBC) = $\frac{1}{2}$ar(ABC)
= $\frac{1}{2}$×$\frac{1}{2}$ar(ABCD) (From (1))
= $\frac{1}{4}$ar(ABCD) ...(2)

Thus, ar(AXCD) = ar(ABCD) − ar(ΔXBC)
⇒ 24 = ar(ABCD) − $\frac{1}{4}$ar(ABCD) (From (2))
⇒ 24 = $\frac{3}{4}$ar(ABCD)
⇒ ar(ABCD) = $24\times \frac{4}{3}$
⇒ ar(ABCD) = 8 × 4
⇒ ar(ABCD) = 32 cm²

From (1)
ar(ΔABC) = $\frac{1}{2}$ar(ABCD)
= $\frac{1}{2}$× 32
= 16 cm²


Hence, ​ar(ΔABC) = 16 cm².