Question

In the adjoining figure, show that ABCD is a parallelogram.
Calculate the area of || gm ABCD.

Answer 1

Given: A quadrilateral ABCD and BD is a diagonal.
To prove: ABCD is a parallelogram.
Construction: Draw AM ⊥ DC and CL ⊥ AB (extend DC and AB). Join AC, the other diagonal of ABCD.

Proof: ar(quad. ABCD) = ar(∆ABD) + ar(​∆DCB)
= 2 ar(​∆ABD) [∵ ar​(∆ABD) = ar(​∆DCB)]
∴ ar(​∆ABD) = $\frac{1}{2}$ar(quad. ABCD) ...(i)

Again, ar(quad. ABCD) = ar(∆ABC) + ar(​∆CDA)
= 2 ar(​∆ ABC) [∵ ar​(∆ABC) = ar(​∆CDA)]
∴ ar(​∆ABC) = $\frac{1}{2}$ar(quad. ABCD) ...(ii)
From (i) and (ii), we have:
ar(​∆ABD) = ar(​∆ABC) = $\frac{1}{2}$ AB ⨯ BD = $\frac{1}{2}$ AB ⨯ CL
⇒ CL = BD
⇒ DC |​​| AB
Similarly, AD |​​| BC.
Hence, ABCD is a paralleogram.
∴ ar(​|​| gm ABCD) = base ​⨯ height = 5 ​⨯ 7 = 35 cm²