Question

The sides of a triangular polt are in the ratio 3:5:7 and its perimeter is 300m. Find its area

Answer 1

given the ratio between the sides of a triangular plot is 3 : 5 : 7 and it's perimeter is given 300m.

let the sides of the triangular plot be 3x, 5x and 7x respectively.

we know that, sum of all sides = perimeter of the triangle

➡ 3x + 5x + 7x = 300m

➡ 15x = 300m

➡ x = 300/15

➡ x = 20m

hence, the sides of the triangle are :-

• 3x = 3 × 20 = 60m

• 5x = 5 × 20 = 100m

• 7x = 7 × 20 = 140m

now, altitude of the triangle is not given neither it's an equilateral triangle.

so we'll use heron's formula to find it's area.

heron's formula = √s(s - a)(s - b)(s - c)

where s is the semi-perimeter of the triangle and a, b and c are the sides of the triangle respectively.

» s = 300/2 = 150m

» a = 60m

» b = 100m

» c = 140m

hence, it's area = √[150(150 - 60)(150 - 100)(150 - 140)]

= √(150 × 90 × 50 × 10)

= √(2 × 3 × 5 × 5 × 2 × 3 × 3 × 5 × 2 × 5 × 5 × 2 × 5)

= √(2 × 2 × 2 × 2 × 3 × 3 × 3 × 5 × 5 × 5 × 5 × 5 × 5)

= 2 × 2 × 3 × 5 × 5 × 5√3

= 1500√3m²

hence the area of the triangular plot is 1500√3m²

Answer 2

given the ratio between the sides of a triangular plot is 3 : 5 : 7 and it's perimeter is given 300m.

let the sides of the triangular plot be 3x, 5x and 7x respectively.

we know that, sum of all sides = perimeter of the triangle

➡ $\large \:3x \:+ \:5x \:+ \:7x \:= \:300m$

➡ $\large \:15x \:= \:300m$

➡ $\large \:x \:= \frac{300}{15}$

➡ $\large \:x \:= \:20m$

hence, the sides of the triangle are :-

• $\large \:3x \:= \:3 \:× \:20 \:= \:60m$

• $\large \:5x \:= \:5 \:× \:20 \:= \:100m$

• $\large \:7x \:= \:7 \:× \:20 \:= \:140m$

now, altitude of the triangle is not given neither it's an equilateral triangle.

so we'll use heron's formula to find it's area.

heron's formula = √s(s - a)(s - b)(s - c)

where s is the semi-perimeter of the triangle and a, b and c are the sides of the triangle respectively.

» $\large \:s \:= \frac{300}{2} \:= \:150m$

» $\large \:a \:= \:60m$

» $\large \:b \:= \:100m$

» $\large \:c \:= \:140m$

hence, it's area = √[150(150 - 60)(150 - 100)(150 - 140)]

= √(150 × 90 × 50 × 10)

= √(2 × 3 × 5 × 5 × 2 × 3 × 3 × 5 × 2 × 5 × 5 × 2 × 5)

= √(2 × 2 × 2 × 2 × 3 × 3 × 3 × 5 × 5 × 5 × 5 × 5 × 5)

= 2 × 2 × 3 × 5 × 5 × 5√3

= 1500√3m²

hence the area of the triangular plot is 1500√3m²