Question:
Sides of a triangle are in the ratio of 12: 17: 25 and its perimeter is 540cm. Find its area.
Sides of a triangle are in the ratio of 12: 17: 25 and its perimeter is 540cm. Find its area.
Updated On: Jun 9, 2024
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Solution and Explanation
Let the common ratio between the sides of the given triangle be x.
Therefore, the side of the triangle will be 12x, 17x, and 25x.
Perimeter of this triangle = 540 cm
12x + 17x + 25x = 540 cm
\(⇒\) 54x = 540
x = \(\frac{540}{54}\)
x = 10 cm
The sides of the triangle:
12x = 12 × 10 = 120 cm,
17x = 17 × 10 = 170 cm,
25x = 25 × 10 = 250 cm
a = 120cm, b = 170 cm, c = 250 cm
Semi-perimeter(s) = \(\frac{540}{2}\) = 270 cm
By Heron’s formula,
Area of a triangle =\( \sqrt{\text{s(s - a)(s - b)(s - c)}}\)
\(= \sqrt{\text{270(270 - 120)(270 - 170)(270 - 250)}}\)
\(= \sqrt{\text{270 × 150 × 100 × 20}}\)
\(= \sqrt{81000000}\)
= 9000 cm2
Area of the triangle = 9000 cm2.
Therefore, the side of the triangle will be 12x, 17x, and 25x.
Perimeter of this triangle = 540 cm
12x + 17x + 25x = 540 cm
\(⇒\) 54x = 540
x = \(\frac{540}{54}\)
x = 10 cm
The sides of the triangle:
12x = 12 × 10 = 120 cm,
17x = 17 × 10 = 170 cm,
25x = 25 × 10 = 250 cm
a = 120cm, b = 170 cm, c = 250 cm
Semi-perimeter(s) = \(\frac{540}{2}\) = 270 cm
By Heron’s formula,
Area of a triangle =\( \sqrt{\text{s(s - a)(s - b)(s - c)}}\)
\(= \sqrt{\text{270(270 - 120)(270 - 170)(270 - 250)}}\)
\(= \sqrt{\text{270 × 150 × 100 × 20}}\)
\(= \sqrt{81000000}\)
= 9000 cm2
Area of the triangle = 9000 cm2.
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