Question

Find the area of the triangle in which

(i) a = 13 m, b = 14 m, c = 15m:
(ii) a = 52 m, b = 56 cm, c = 60 cm:
(iii) a = 91 m, b = 98 m, c = 105 m.

Answer 1

(i) Let a = 13 m, b = 14 m and c = 15 m
s = $\left(\frac{a+b+c}{2}\right)$ = $\left(\frac{13+14+15}{2}\right)=\left(\frac{42}{2}\right)$m = 21 m
∴ Area of the triangle = $\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$ sq. units
= $\sqrt{21\left(21-13\right)\left(21-14\right)\left(21-15\right)}$m²
= $\sqrt{21\times 8\times 7\times 6}$ m²
= $\sqrt{3\times 7\times 2\times 2\times 2\times 7\times 2\times 3}$ m²
= (2 $\times$ 2 $\times$ 3 $\times$ 7) m²
= 84 m²

(ii) Let a = 52 cm, b = 56 cm and c = 60 cm
s = $\left(\frac{a+b+c}{2}\right)$ = $\left(\frac{52+56+60}{2}\right)=\left(\frac{168}{2}\right)$ cm = 84 cm
∴ Area of the triangle = $\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$ sq. units
= $\sqrt{84\left(84-52\right)\left(84-56\right)\left(84-60\right)}$cm²
= $\sqrt{84\times 32\times 28\times 24}$ cm²
= $\sqrt{12\times 7\times 4\times 8\times 4\times 7\times 3\times 8}$ cm²
= $\sqrt{2\times 2\times 3\times 7\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 7\times 3\times 2\times 2\times 2}$ cm²
= (2 $\times$ 2 $\times$ 2 $\times$ 2 $\times$ 2 $\times$ 2 $\times$ 3 $\times$ 3 $\times$ 7) m²
= 1344 cm²

(iii) Let a = 91 m, b = 98 m and c = 105 m
s = $\left(\frac{a+b+c}{2}\right)$ = $\left(\frac{91+98+105}{2}\right)=\left(\frac{294}{2}\right)$ m = 147 m
∴ Area of the triangle = $\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$ sq. units
= $\sqrt{147\left(147-91\right)\left(147-98\right)\left(147-105\right)}$m²
= $\sqrt{147\times 56\times 49\times 42}$ m²
= $\sqrt{3\times 49\times 8\times 7\times 49\times 6\times 7}$ m²
= $\sqrt{3\times 7\times 7\times 2\times 2\times 2\times 7\times 7\times 7\times 2\times 3\times 7}$ m²
= ( 2 $\times$ 2 $\times$ 3 $\times$ 7 $\times$ 7 $\times$ 7) m²
= 4116 m²