Question

The perimeter of a traingle is 300cm.If it's sides are in the ratio 3:5:7. Find the area of the triangle and the height corresponding to the shortest side.

Answer 1

$\bold{Given}$

The perimeter of a triangle is 300 cm.

And its sides are in ratio 3 : 5 : 7.

Let common ratio = x

So, sides become = 3x, 5x and 7x

Perimeter = sum of all sides.

$\mathsf{300 = 3x + 5x + 7x}$

$\mathsf{300 = 15x}$

$\mathsf{\frac{300}{15} = x}$

$\mathsf{20= x}$

So, sides are,

$\mathsf{3 \times 20= 60}$

$\mathsf{7 \times 20= 140}$

$\mathsf{5 \times 20= 100}$

Since all sides are unequal therefore, its a scalene triangle,

So for area ,

semi perimeter s = $\frac{s_1 + s_2 + s_3}{2}$

semi perimeter s = $\frac{300}{2}$

semi perimeter s = 150 cm

For area ,

$\mathsf{A = \sqrt{s \times (s-a) (s-b) (s-c)}}$

$\mathsf{A = \sqrt{150 \times (150 - 60)(150-100)(150-140)}}$

$\mathsf{A = \sqrt{150 \times (90)(50)(10)}}$

$\mathsf{A = \sqrt{5 \times 10 \times 3 \times 3 \times 3 \times 10 \times 5 \times 10 \times 10}}$

$\mathsf{A = \sqrt{5^{2} + 3^{3} + 10^{4}}}$

$\mathsf{A = \sqrt{5 + 3^{3} + 10^{2}}}$

$\mathsf{A =1500\sqrt{3}}$

Let the triangle be ABC so side

AB as shortest side = 60 cm

BC longest side = 140 cm

CA = 100 cm.