Class 9 Exam  >  Class 9 Notes  >  RD Sharma Solutions for Class 9 Mathematics  >  RD Sharma Solutions Ex-12.1, Heron's Formula, Class 9, Maths

RD Sharma Solutions Ex-12.1, Heron's Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics PDF Download

Q1 . Find  the area of a triangle whose sides are respectively 150 cm, 120 cm and 200 cm.

Solution:

Let the sides of the given triangle be a, b, c respectively.

So given,

a = 150 cm

b = 120 cm

c = 200 cm

By using Heron’s Formula

The area of the triangle =  RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

Semi perimeter of a triangle = s

2s = a + b + c

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

s = 235 cm

Therefore, area of the triangle  RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics
= 8966.56cm2


Q2. Find  the area of a triangle whose sides are respectively 9 cm, 12 cm and 15 cm.

Solution:

Let the sides of the given triangle be a, b, c respectively.

So given,

a = 9 cm

b = 12 cm

c = 15 cm

By using Heron’s Formula

The area of the triangle =  RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

Semi perimeter of a triangle = s

2s = a + b + c

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

s = 18 cm

Therefore, area of the triangle   = RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

= 54cm2


Q3. Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42cm.

Solution:

Whenever we are given the measurements of all sides of a triangle, we basically look for Heron’s formula to find out the area of the triangle.

If we denote area of the triangle by A, then the area of a triangle having sides a, b, c and s as semi-perimeter is given by:

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

We are given:

a = 18 cm

b = 10 cm, and perimeter = 42 cm

We know that perimeter = 2s,

So, 2s = 42

Therefore, s = 21 cm

We know that,  RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

42 = 28 + c

c = 14 cm

So the area of the triangle is:

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics


Q4 . In a triangle ABC, AB = 15cm, BC = 13cm and AC = 14cm. Find the area of triangle ABC and hence its altitude on AC.

Solution:

Let the sides of the given triangle be AB = a, BC = b, AC =  c respectively.

So given,

a = 15 cm

b = 13 cm

c = 14 cm

By using Heron’s Formula

The area of the triangle  =RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

Semi perimeter of a triangle= 2s

2s = a + b + c

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

s = 21 cm

Therefore, area of the triangle RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

= 84cm2

BE is a perpendicular on AC

Now, area of triangle = 84cm2

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

BE = 12cm

The length of BE is 12 cm


Q5 . The perimeter of a triangular field is 540 m and its sides are in the ratio 25:17:12. Find the area of triangle.

Solution:

Let the sides of the given triangle be a = 25x, b = 17x, c = 12x respectively,

So,

a = 25x cm

b = 17x cm

c = 12x cm

Given Perimeter = 540 cm

2s = a + b + c

a + b + c = 540 cm

25x + 17x + 12x = 540 cm

54x = 540 cm

x = 10 cm

Therefore, the sides of a triangle are

a = 250 cm

b = 170 cm

c = 120 cm

Now, Semi perimeter  RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

= 540/2

= 270 cm

By using Heron’s Formula

The area of the triangle RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

= 9000cm2

Therefore, the area of the triangle is 9000cm2


Q6. The perimeter of a triangle is 300 m. If its sides are in the ratio of 3: 5: 7. Find the area of the triangle.

Solution:

Given the perimeter of a triangle is 300 m and the sides are in a ratio of 3: 5: 7

Let the sides a, b, c of a triangle be 3x, 5x, 7x respectively

So, the perimeter = 2s = a + b + c

200 = a + b + c

300 = 3x + 5x + 7x

300 = 15x

Therefore, x = 20 m

So, the respective sides are

a = 60 m

b = 100 m

c = 140 m

Now, semi perimeter  RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

= 150 m

By using Heron’s Formula

The area of a triangle  RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

Thus, the area of a triangle is  RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics


Q7. The perimeter of a triangular field is 240 dm. If two of its sides are 78 dm and 50 dm, find the length of the perpendicular on the side of length 50 dm from the opposite vertex.

Solution:

Given,

In a triangle ABC, a = 78 dm = AB, b = 50 dm = BC

Now, Perimeter = 240 dm

Then, AB + BC + AC = 240 dm

78 + 50 + AC = 240

AC = 240-(78+50)

AC = 112 dm = c

Now, 2s = a + b + c

2s = 78 + 50 + 112

s = 120 dm

Area of a triangle ABC RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

= 1680dm2

Let AD be a perpendicular on BC

Area of the triangle ABC =  RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

AD = 67.2 dm


Q8. A triangle has sides 35 cm, 54 cm, 61 cm long. Find its area. Also, find the smallest of its altitudes?

Solution:

Given,

The sides of the triangle are

a = 35 cm

b = 54 cm

c = 61 cm

Perimeter 2s = a + b + c

2s = 35 + 54 + 61 cm

Semi perimeter s = 75 cm

By using Heron’s Formula,

Area of the triangle  RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

= 939.14cm2

The altitude will be smallest provided the side corresponding to this altitude is longest.

The longest side = 61 cm

Area of the triangle  RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

h = 30.79cm

Hence the length of the smallest altitude is 30.79cm


Q9.The lengths of the sides of a triangle are in a ratio of 3 : 4 : 5 and its perimeter is 144 cm. Find the area of the triangle and the height corresponding to the longest side?

Solution:

Given the perimeter of a triangle is 160m and the sides are in a ratio of  3 : 4 : 5

Let the sides a, b, c of a triangle be 3x, 4x, 5x respectively

So, the perimeter = 2s = a + b + c

144 = a + b + c

144 = 3x+ 4x+ 5x

Therefore, x = 12cm

So, the respective sides are

a = 36cm

b = 48cm

c = 60cm

Now, semi perimeter  RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

= 72 cm

By using Heron’s Formula

The area of a triangle RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

= 864cm2

Thus, the area of a triangle is 864cm2

The altitude will be smallest provided the side corresponding to this altitude is longest.

The longest side = 60 cm

Area of the triangle RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

h = 28.8 cm

Hence the length of the smallest altitude is 28.8 cm


Q10. The perimeter of an isosceles triangle is 42 cm and its base is 3/2 times each of the equal side. Find the length of each of the triangle, area of the triangle and the height of the triangle.

Solution:

Let ‘x’ be the length of two equal sides,

Therefore the base  RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

Let the sides a, b, c of a triangle be RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics  respectively

So, the perimeter = 2s = a + b + c

42 = a + b + c

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

Therefore, x = 12 cm

So, the respective sides are

a = 12 cm

b = 12 cm

c = 18 cm

Now, semi perimeter RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

= 21 cm

By using Heron’s Formula

The area of a triangle RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

= 71.42cm2

Thus, the area of a triangle is 71.42cm2

The altitude will be smallest provided the side corresponding to this altitude is longest.

The longest side = 18 cm

Area of the triangle RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

h = 7.93 cm

Hence the length of the smallest altitude is 7.93 cm


Q11. Find the area of the shaded region in fig. below

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

Solution:

Area of the shaded region = Area of ΔABC−Area of ΔADB

Now in triangle ADB

AB2 = AD2+BD2………(i)

Given, AD = 12 cm, BD =16 cm

Substituting the value of AD and BD in eq (i), we get

AB2 = 122+162

= 400cm2

AB = 20 cm

Now, area of a triangle RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

= 96cm2

Now in triangle ABC,

The area of a triangle RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

RD Sharma Solutions Ex-12.1, Heron`s Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics

= 480cm2

Thus, the area of a triangle is 480cm2

Area of shaded region = Area of triangle ABC – Area of triangle ADB

= (480 – 96) cm2

= 384 cm2

Area of shaded region = 384 cm2

The document RD Sharma Solutions Ex-12.1, Heron's Formula, Class 9, Maths | RD Sharma Solutions for Class 9 Mathematics is a part of the Class 9 Course RD Sharma Solutions for Class 9 Mathematics.
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FAQs on RD Sharma Solutions Ex-12.1, Heron's Formula, Class 9, Maths - RD Sharma Solutions for Class 9 Mathematics

1. What is Heron's formula and how is it used to find the area of a triangle?
Ans. Heron's formula is used to find the area of a triangle when the lengths of all three sides are known. The formula is given as: Area = √[s(s-a)(s-b)(s-c)] where s is the semi-perimeter of the triangle, and a, b, and c are the lengths of the three sides of the triangle. To find the area using Heron's formula, we first calculate the semi-perimeter by adding all three sides and dividing the sum by 2. Then, we substitute the values of the semi-perimeter and the side lengths into the formula to calculate the area.
2. How do you calculate the semi-perimeter of a triangle?
Ans. The semi-perimeter of a triangle is calculated by adding the lengths of all three sides and dividing the sum by 2. Mathematically, it can be expressed as: Semi-perimeter = (a + b + c) / 2 where a, b, and c are the lengths of the three sides of the triangle. The semi-perimeter is used in Heron's formula to find the area of the triangle.
3. Can Heron's formula be used for any type of triangle?
Ans. Yes, Heron's formula can be used to find the area of any type of triangle, whether it is equilateral, isosceles, or scalene. The formula only requires the lengths of the three sides of the triangle to calculate the area, regardless of the triangle's shape or angles.
4. Can Heron's formula be used to find the area of a triangle if only the side lengths are given?
Ans. Yes, Heron's formula can be used to find the area of a triangle if only the side lengths are given. The formula does not require any angle measurements, so it can be used solely based on the lengths of the sides. By substituting the side lengths into the formula, the area of the triangle can be calculated.
5. Are there any other formulas to find the area of a triangle apart from Heron's formula?
Ans. Yes, apart from Heron's formula, there are other formulas to find the area of a triangle. For example, if the base and height of the triangle are known, the area can be calculated using the formula: Area = (1/2) * base * height where base is the length of the base of the triangle and height is the perpendicular distance from the base to the opposite vertex. This formula is commonly used for right-angled triangles. Additionally, if the lengths of two sides and the included angle are known, the area can be calculated using the formula: Area = (1/2) * a * b * sin(C) where a and b are the lengths of the two sides, and C is the included angle between them.
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