Question

Assertion: The sides of a ∆ABC are in the ratio 2 : 3 : 4 and its perimeter is 36 cm. Then, $\mathrm{ar}(∆ABC)=12\sqrt{15}{\mathrm{cm}}^2$.
Reason: If 2s = (a + b + c), where a, b, c are the sides of a triangle, then its area $=\sqrt{(s-a)(s-b)(s-c)}$.
(a) Both Assertion and Reason are true and Reason is a correct explanation of Assertion.
(b) Both Assertion and Reason are true but Reason is not a correct explanation of Assertion.
(c) Assertion is true and Reason is false.
(d) Assertion is false and Reason is true.

Answer 1

(c) Assertion is true and Reason is false.
Assertion:
Let the sides of the triangle be 2x cm, 3x cm and 4x cm.
Perimeter = Sum of all sides
or, 36 = 2x + 3x + 4x
or, 9x = 36
or, x = 4
Thus, the sides of the triangle are 2$\times$4 cm, 3$\times$4 cm and 4$\times$4 cm, i.e., 8 cm, 12 cm and 16 cm.

Now,
$$\begin{gathered} \mathrm{Let}: \\ a=8\mathrm{cm},b=12\mathrm{cm}\mathrm{and}c=16\mathrm{cm} \\ s=\frac{36}{2}=18\mathrm{cm} \\ \mathrm{By}\mathrm{Heron}\text{'}\mathrm{s}\mathrm{formula},\mathrm{we}\mathrm{have}: \\ \mathrm{Area}\mathrm{of}\mathrm{triangle}=\sqrt{s(s-a)(s-b)(s-c)} \\ =\sqrt{18(18-8)(18-12)(18-16)} \\ =\sqrt{18\times 10\times 6\times 2} \\ =\sqrt{6\times 3\times 5\times 2\times 6\times 2} \\ =6\times 2\sqrt{15} \\ =12\sqrt{15}{\mathrm{cm}}^2 \end{gathered}$$
Hence, Assertion is true.
Reason: If 2s = (a + b + c), where a, b and c are the sides of a triangle, then its area $=\sqrt{(s-a)(s-b)(s-c)}$.
Hence, it is false.

Area should be $\sqrt{s(s-a)(s-b)(s-c)}$.