Question

In a quadrilateral ABCD, AB = 28 cm, BC = 26 cm, CD = 50 cm, DA = 40 cm and diagonal AC = 30 cm. Find the area of the quadrilateral.

Answer 1

First, we have to find the area of ΔABC and ΔACD.
For ΔACD:
Let a = 30 cm, b = 40 cm and c = 50 cm
s = $\left(\frac{a+b+c}{2}\right)=\left(\frac{30+40+50}{2}\right)=\left(\frac{120}{2}\right)=60$ cm
∴ Area of triangle ACD = $\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$ sq. units
= $\sqrt{60\left(60-30\right)\left(60-40\right)\left(60-50\right)}$ cm²
= $\sqrt{60\times 30\times 20\times 10}$ cm²
= $\sqrt{360000}$ cm²
= 600 cm²
For ΔABC:
Let a = 26 cm, b = 28 cm and c = 30 cm
s = $\left(\frac{a+b+c}{2}\right)=\left(\frac{26+28+30}{2}\right)=\left(\frac{84}{2}\right)=42$ cm
∴ Area of triangle ABC = $\sqrt{s\left(s-a\right)\left(s-b\left(s-c\right)\right)}$ sq. units
= $\sqrt{42\left(42-26\right)\left(42-28\right)\left(42-30\right)}$ cm²
= $\sqrt{42\times 16\times 14\times 12}$ cm²
= $\sqrt{2\times 3\times 7\times 2\times 2\times 2\times 2\times 2\times 7\times 3\times 2\times 2}$ cm²
= (2 $\times$2 $\times$ 2 $\times$ 2 $\times$ 3 $\times$ 7) cm²
= 336 cm²
∴ Area of the given quadrilateral ABCD = Area of ΔACD + Area of ΔABC
= (600 + 336) cm² = 936 cm²