NCERT Class 9 Maths Solutions: Heron’s Formula

Question:

To find the area of a triangle when all side lengths are known, Heron’s formula is used. For an isosceles triangle, two sides are equal. We can find the length of the third side using the perimeter.


1. Identify the given information:
Perimeter of the isosceles triangle = 30 cm
Length of each of the equal sides = 12 cm

2. Find the length of the third side:
Let the lengths of the sides of the isosceles triangle be a, b, and c.
Since it’s an isosceles triangle, two sides are equal. Let a = 12 cm and b = 12 cm.
The perimeter is the sum of all sides: Perimeter = a + b + c
30 cm = 12 cm + 12 cm + c
30 cm = 24 cm + c
c = 30 cm – 24 cm
c = 6 cm
So, the sides of the triangle are 12 cm, 12 cm, and 6 cm.

3. Calculate the semi-perimeter (s):
The semi-perimeter is half of the perimeter.
s = Perimeter / 2
s = 30 cm / 2
s = 15 cm

4. Apply Heron’s Formula to find the area:
Heron’s formula for the area of a triangle with sides a, b, and c and semi-perimeter s is:
Area = sqrt(s * (s – a) * (s – b) * (s – c))

Substitute the values:
Area = sqrt(15 * (15 – 12) * (15 – 12) * (15 – 6))
Area = sqrt(15 * 3 * 3 * 9)
Area = sqrt(15 * 9 * 9)
Area = sqrt(15 * 81)
Area = sqrt(1215)

5. Simplify the square root:
We can factor 1215 to simplify the square root.
1215 = 5 * 243 = 5 * 3 * 81 = 5 * 3 * 9 * 9 = 5 * 3 * 3^2 * 3^2 = 15 * 81
Area = sqrt(81 * 15)
Area = sqrt(81) * sqrt(15)
Area = 9 * sqrt(15) cm^2

The area of the triangle is 9 * sqrt(15) cm^2.

Question:

To find the area of a triangle when sides are in a given ratio and perimeter is known, we first find the actual lengths of the sides using the perimeter. Then, we use Heron’s formula to calculate the area. Heron’s formula states that the area of a triangle with sides a, b, and c is given by Area = sqrt(s(s-a)(s-b)(s-c)), where s is the semi-perimeter (s = (a+b+c)/2).


Let the sides of the triangle be 12x, 17x, and 25x, where x is a common multiplier.
The perimeter of the triangle is given as 540 cm.
Perimeter = Sum of all sides
540 = 12x + 17x + 25x
540 = 54x
x = 540 / 54
x = 10 cm

Now, we find the actual lengths of the sides:
Side a = 12x = 12 * 10 = 120 cm
Side b = 17x = 17 * 10 = 170 cm
Side c = 25x = 25 * 10 = 250 cm

Next, we calculate the semi-perimeter (s):
s = Perimeter / 2
s = 540 / 2
s = 270 cm

Now, we use Heron’s formula to find the area of the triangle:
Area = sqrt(s(s-a)(s-b)(s-c))
Area = sqrt(270 * (270 – 120) * (270 – 170) * (270 – 250))
Area = sqrt(270 * 150 * 100 * 20)

To simplify the calculation:
Area = sqrt((27 * 10) * (15 * 10) * 100 * (2 * 10))
Area = sqrt(27 * 15 * 2 * 10 * 10 * 100 * 10)
Area = sqrt(27 * 15 * 2 * 100000)
Area = sqrt(27 * 30 * 100000)
Area = sqrt(810 * 100000)
Area = sqrt(81 * 10 * 100000)
Area = sqrt(81 * 1000000)
Area = sqrt(81) * sqrt(1000000)
Area = 9 * 1000
Area = 9000 sq cm