Sides of a triangle are in a ratio of 12:17:25 and its perimeter is 540 cm. Find its area by heron's formula?

Question

Answer

Perimeter= 540cm ; a:b:c= 12:17:25
Let, a= 12x
b= 17x
c= 25x
12x+17x+25x=540
54x=540
x= 540/54
x= 10
Then, a= 12x=12×10=120cm
b= 17x=17×10=170cm
c= 25x=25×10=250cm
By HERON'S FORMULA,
S ( semi perimeter )= a+b+c/2
= 540/2
= 270cm
Area of triangle=√ s (s-a)(s-b)(s-c)
=√270 (270-120)(270-170)(270-250)
=√270×150×100×20
=√2×3×3×3×5×2×3×5×5×2×2×5×5×2×2×5
= 2×3×3×5×3×5×2×5×2
= 9000cm ^2..

Answer

Let the common multiple be x
a=12x
b= 17x
c = 25x
perimeter =540 cm
perimeter = a+b+c
540 = 12x+17x 25x
540 = 54x
x = 10 cm

a=120 cm
b= 170 cm
c= 250 cm

semiperimeter = a+b+c/2
= 540/2
= 270 cm

area = 9000 cm²

Answer

Given:
The sides of the triangle are in the ratio of 12:17:25.
The perimeter of the triangle is 540 cm.

To Find:
The area of the triangle using Heron's formula.

Explanation:
Step 1: Finding the Lengths of the Sides
Let's assume the lengths of the sides of the triangle are 12x, 17x, and 25x, where x is a common factor.

According to the given ratio, the sum of these three sides is equal to the perimeter of the triangle:
12x + 17x + 25x = 540
54x = 540
x = 10

So, the lengths of the sides are:
12x = 12 * 10 = 120 cm
17x = 17 * 10 = 170 cm
25x = 25 * 10 = 250 cm

Step 2: Calculating the Semi-perimeter
The semi-perimeter of a triangle is half of its perimeter. Therefore, the semi-perimeter is:
S = (120 + 170 + 250) / 2 = 540 / 2 = 270 cm

Step 3: Applying Heron's Formula
Heron's formula states that the area (A) of a triangle with sides a, b, and c is given by:
A = √(s(s-a)(s-b)(s-c))

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

Substituting the values into the formula, we have:
A = √(270(270-120)(270-170)(270-250))

Step 4: Simplifying the Expression
Using the values in the formula, we get:
A = √(270 * 150 * 100 * 20)

Step 5: Calculating the Area
Simplifying the expression further, we have:
A = √(81000000) = 9000 cm²

Therefore, the area of the triangle is 9000 cm².

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