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.

Similar Class 9 Doubts

Sides of a triangle are in the ratio of 12:17:25 and its perimeter is 540cm find its area?

the perimeter of triangle is 12cm and its hypotenuse is 5cm find the both sides and area of triangle verify the result obtained by heron's formula?

find a,b,c if the perimeter of a triangle is 36 cm and it's side's are in the ratio a:b:c=3:4:5 ,heron's formula

The sides of a triangle are in the ratio of 2:3:4 and its perimeter is 90cm.find it's area.?

A traffic signal board indicating School Ahead is equilateral triangle with side 'a' find the area of the signal board ,using Heron 's formula .if its perimeter is 180cm,what will be the area of signal board?

Top Courses for Class 9View all

Science Class 9

Social Studies (SST) Class 9

Mathematics (Maths) Class 9

English Grammar Advanced

English Class 9

Top Courses for Class 9

Top Courses for Class 9

Suggested Free Tests

Related Content

Class 9 Study Material

Class 9 Subjects

Competitive Examinations for Class 9

NCERT Books

NCERT Solutions

Reference Books for Class 9

Other Popular Exams

Company