The perimeter of an iscosceles triangle is 42 cm and its base is [3/2] times each of the equal sides - Maths - Heron\s Formula

Question

The perimeter of an iscosceles triangle is 42 cm and its base is
[3/2] times each of the equal sides Find the length of each side of
the triangle,area of the triangle and the height of the
triangle

Ankush Jain · Accepted Answer

Given, perimeter of an isosceles triangle = 42 cm

Let the equal side of triangle be x cm

Base of triangle = $\frac{3}{2}$ x

So, perimeter = 42 cm

$\Rightarrow x + x + \frac{3}{2} x = 42 \Rightarrow 2 x + \frac{3 x}{2} = 42 \Rightarrow \frac{7 x}{2} = 42 \Rightarrow x = 12 cm ∴ base of triangle = \frac{3}{2} x = \frac{3}{2} \times 12 cm = 18 cm$

So, length of each side of the triangle is 12cm, 12cm and 18cm respectively.

Now,

$Area of Δ = \sqrt{s (s - a) (s - b) (s - c)} \Rightarrow s = \frac{a + b + c}{2} = \frac{42}{2} = 21 \Rightarrow Area of Δ = \sqrt{21 (21 - 12) (21 - 12) (21 - 18)} = \sqrt{21 \times 9 \times 9 \times 3} = \sqrt{3 \times 7 \times 9 \times 9 \times 3} = 9 \times 3 \sqrt{7} = 27 \sqrt{7} {cm}^{2} ...... (1) Also, Area of Δ = \frac{1}{2} \times base \times height = \frac{1}{2} \times 18 \times h = 9 h ..... (2)$

Equating (1) and (2), we get

$\Rightarrow 9 h = 27 \sqrt{7} \Rightarrow h = 3 \sqrt{7} cm$

The perimeter of an iscosceles triangle is 42 cm and its base is [3/2] times each of the equal sides.Find the length of each side of the triangle,area of the triangle and the height of the triangle.