NCERT Solutions for Class 9 Maths Chapter 10
Heron’s formula It is particularly useful when it is not possible to find the height of the triangle easily.
The Formula:
Area = √s(s – a)(s – b)(s – c)
Where
s = (a + b + c)/2
Key Points
- Semi-perimeter: Half the perimeter of a triangle.
- Application: Can be used to find the area of any triangle, regardless of its shape.
- Derivation: The formula can be derived using the Pythagorean theorem and the properties of triangles.
In essence, Heron’s formula provides a direct method to calculate the area of a triangle when all side lengths are known, without requiring the calculation of height or other trigonometric functions.
NCERT Solutions for Class 9 Maths Chapter 10
Exercise 10.1
1. A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side a. Find the area of the signal board, using Heron’s formula.If its perimeter is 180 cm, what will be the area of the signal board?
Ans :
Part 1: Using Heron’s Formula
- For an equilateral triangle,
- all sides are equal, so a = b = c.
- Semi-perimeter (s) = (a + b + c)/2 = (a + a + a)/2 = 3a/2
- Using Heron’s formula: Area = √[s(s – a)(s – b)(s – c)] = √[(3a/2)(3a/2 – a)(3a/2 – a)(3a/2 – a)] = √[(3a/2)(a/2)(a/2)(a/2)] = (a²/4)√3
Therefore, the area of the equilateral triangle is (a²/4)√3 square units.
Part 2: Given Perimeter is 180 cm
- Perimeter of an equilateral triangle = 3a
- So, 3a = 180 cm
- a = 60 cm
Substituting a = 60 cm in the area formula:
- Area = (60²/4)√3 = 900√3 cm²
Therefore, the area of the traffic signal board is 900√3 cm².
2. The triangular side walls of a flyover have been used for advertisements. The sides of the walls are 122 m, 22 m and 120 m (see figure). The advertisements yield an earning of ₹5000 per m² per year. A company hired one of its walls for 3 months. How much rent did it pay?
Ans :
Step 1: Calculate the area of the triangular wall
Semi-perimeter (s) = (a + b + c)/2 = (122 + 22 + 120)/2 = 132 m
- Area = √[s(s – a)(s – b)(s – c)] = √[132(132-122)(132-22)(132-120)] = √[13210110*12] = 1210 m²
Step 2: Calculate the rent for the entire year
Rent per m² per year = ₹5000
Total area = 1210 m²
Rent for the entire year = 5000 * 1210 = ₹6050000
Step 3: Calculate the rent for 3 months
Rent for 3 months = (Rent for entire year) * (3/12)
= 6050000 * (1/4)
= ₹1512500
Therefore, the company paid ₹1512500 as rent.
3. There is a slide in a park. One of its side Company hired one of its walls for 3 months.walls has been painted in some colour with a message “KEEP THE PARK GREEN AND CLEAN” (see figure). If the sides of the wall are 15 m, 11 m and 6m, find the area painted in colour.
Ans :
Since the wall is triangular, we can use Heron’s formula to find its area.
Heron’s formula:
- Area of a triangle = √(s(s-a)(s-b)(s-c)) Where:
- s is the semi-perimeter (half the perimeter)
Calculations:
- Calculate the semi-perimeter (s):
- s = (a + b + c) / 2 = (15 + 11 + 6) / 2 = 32 / 2 = 16 m
- Calculate the area using Heron’s formula:
- Area = √(s(s-a)(s-b)(s-c))
- = √(16 * (16-15) * (16-11) * (16-6))
- = √(16 * 1 * 5 * 10)
- = √800 = 20√2 square meters
4. Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm.
Ans :
Step 1: Find the third side.
- Let the third side be ‘c’.
- Perimeter = a + b + c
- 42 = 18 + 10 + c
- c = 42 – 18 – 10 = 14 cm
Step 2: Use Heron’s formula
- Area = √(s(s-a)(s-b)(s-c)) Where s is the semi-perimeter, calculated as s = (a + b + c) / 2
- Calculate the semi-perimeter:
- s = (18 + 10 + 14) / 2 = 21 cm
- Calculate the area:
- Area = √(21 * (21-18) * (21-10) * (21-14)) = √(21 * 3 * 11 * 7) = √(3 * 7 * 3 * 11 * 7) = 3 * 7 * √11 = 21√11 cm²
Therefore, the area of the triangle is 21√11 square centimeters.
5. Sides of a triangle are in the ratio of 12 : 17 : 25 and its perimeter is 540 cm. Find its area.
Ans :
Step 1: Find the sides
Let the common ratio of the sides be ‘x’. So, the sides are 12x, 17x, and 25x cm.
Perimeter = 12x + 17x + 25x = 540 cm => 54x = 540 cm => x = 10 cm
Therefore, the sides of the triangle are:
- a = 12x = 12 * 10 = 120 cm
- b = 17x = 17 * 10 = 170 cm
- c = 25x = 25 * 10 = 250 cm
Step 2: Calculate the semi-perimeter
Semi-perimeter (s) = (a + b + c) / 2 = (120 + 170 + 250) / 2 = 540 / 2 = 270 cm
Step 3: Use Heron’s formula
Area = √(s(s-a)(s-b)(s-c)) =
√(270 * (270-120) * (270-170) * (270-250))
= √(270 * 150 * 100 * 20)
= √(3 * 3 * 3 * 10 * 5 * 2 * 5 * 5 * 2 * 2 * 5 * 2)
= 3 * 3 * 5 * 5 * 2 * 2 * √10
= 900 √10 cm²
Therefore, the area of the triangle is 900√10 square centimeters.
6. An isosceles triangle has perimeter 30 cm and each of the equal sides is 12 cm. Find the area of the triangle.
Ans :
Step 1: Find the length of the third side
- Let the third side be ‘c’.
- Perimeter = a + b + c
- 30 = 12 + 12 + c
- c = 30 – 24 = 6 cm
Step 2: Use Heron’s
- Area = √(s(s-a)(s-b)(s-c)) Where s is the semi-perimeter, calculated as s = (a + b + c) / 2
- Calculate the semi-perimeter:
- s = (12 + 12 + 6) / 2 = 15 cm
- Calculate the area:
- Area = √(15 * (15-12) * (15-12) * (15-6))
- = √(15 * 3 * 3 * 9) = √1215 = 9√15 cm²
Therefore, the area of the isosceles triangle is 9√15 square centimeters.
NCERT Solutions for Class 9 Maths Chapter 10
FAQ’s
What is NCERT Solutions for Class 9 Maths Chapter 10 Heron’s formula and when is it used?
Heron’s formula is used to find the area of a triangle when the lengths of all three sides are known. It is especially useful when the height of the triangle is not easily measurable.
What is NCERT Solutions for Class 9 Maths Chapter 10 the formula for Heron’s theorem?
The formula is:
Area = √[s(s – a)(s – b)(s – c)],
where s = (a + b + c)/2 is the semi-perimeter of the triangle, and a, b, c are the sides.
NCERT Solutions for Class 9 Maths Chapter 10 Can Heron’s formula be used for all types of triangles?
Yes, Heron’s formula can be applied to any triangle — whether it is scalene, isosceles, or equilateral.
What is NCERT Solutions for Class 9 Maths Chapter 10 the area of an equilateral triangle with a perimeter of 180 cm using Heron’s formula?
If the perimeter is 180 cm, each side is 60 cm.
Using Heron’s formula, the area = 900√3 cm².
How NCERT Solutions for Class 9 Maths Chapter 10 is the semi-perimeter (s) calculated in Heron’s formula?
The semi-perimeter is half of the triangle’s perimeter, calculated as:
s = (a + b + c) / 2,
where a, b, and c are the three sides of the triangle.
