Question

The lengths of sides of a triangle are in the ratio 3:4:5 and its perimeter is 144 cm. Find (i) the area of the triangle (ii) the height corresponding to largest side.

Answer 1

Let the first side (a) of the triangle be 3x m, second side (b) be 4x m and third side (c) be 5x m.
∴ Perimeter of the triangle = a + b + c
$\Rightarrow$144 = 3x + 4x + 5x
$\Rightarrow$144 = 12x
$\Rightarrow$x = 12
Now, we have:
a = 3x = 3 × 12 cm = 36 cm
b = 4x = 4 × 12 cm = 48 cm
c = 5x = 5 × 12 cm = 60 cm

(i)Semiperimeter (s) of the triangle = $\frac{144}{2}=72\mathrm{cm}$
Thus, by Heron’s formula, we have:

$$\begin{gathered} \mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{triangle}=\sqrt{72(72-36)(72-48)(72-60)} \\ =\sqrt{72\times 36\times 24\times 12} \\ =\sqrt{36\times 2\times 36\times 12\times 2\times 12} \\ =864{\mathrm{cm}}^2 \end{gathered}$$

(ii) Longest side of the triangle = 60 cm
Lets take this side as the base of the triangle.
Let the height corresponding to the longest side be h cm.
$$\begin{gathered} \mathrm{Then}\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{triangle}=\frac{1}{2}\times \mathrm{base}\times \mathrm{height} \\ \Rightarrow 864=\frac{1}{2}\times 60\times h \\ \Rightarrow 864=30\times h \\ \Rightarrow h=28.8\mathrm{cm} \end{gathered}$$

∴ Height corresponding to the longest side = 28.8 cm