Class 9 Maths Chapter 10 Question Answers - Heron`s Formula

 Page 1


        
     
                                    
            
               
                                    
                
 
 
                     
             
 
 
 
 
        
Question:26
Find the area of a triangle whose base and altitude are 5 cm and 4 cm respectively.
Solution:
Given, base = 5 cm; height = 4 cm
Area of the triangle = 
1
2
×Base ×Height
Question:27
Find the area of a triangle whose sides are 3 cm, 4 cm and 5 cm respectively.
Solution:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
, where
Therefore the area of a triangle, say having sides 3 cm, 4 cm and 5 cm is given by
a = 3 cm ; b = 4 cm ; c = 5 cm
Now, area 
Page 2


        
     
                                    
            
               
                                    
                
 
 
                     
             
 
 
 
 
        
Question:26
Find the area of a triangle whose base and altitude are 5 cm and 4 cm respectively.
Solution:
Given, base = 5 cm; height = 4 cm
Area of the triangle = 
1
2
×Base ×Height
Question:27
Find the area of a triangle whose sides are 3 cm, 4 cm and 5 cm respectively.
Solution:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
, where
Therefore the area of a triangle, say having sides 3 cm, 4 cm and 5 cm is given by
a = 3 cm ; b = 4 cm ; c = 5 cm
Now, area 
= v 6(6 -3)(6 -4)(6 -5) = v 6 ×3 ×2 ×1 = v 36 = 6 cm
2
Question:28
Find the area of an isosceles triangle having the base x cm and one side y cm.
Solution:
Let us assume triangle ABC be the given isosceles triangle having sides AB = AC and base BC. The area of a triangle ABC, say A having given sides AB and AC equals to y cm and
given base BC equals to x cm is given by
Where,
Base = BC = x cm; Height = 
y
2
-
x
2
4
A =
1
2
(Base ×Height) =
1
2
×x y
2
-
x
2
4
=
x
2
y
2
-
x
2
4
Question:29
Find the area of an equilateral triangle having each side 4 cm.
Solution:
Area of an equilateral triangle having each side a cm is given by 
Area of the given equilateral triangle having each equal side equal to 4 cm is given by
a = 4 cm
Question:30
Find the area of an equilateral triangle having each side x cm.
Solution:
Area of an equilateral triangle, say A having each side a cm is given by 
Area of the given equilateral triangle having each equal side equal to x cm is given by
a = x cm
Question:31
The perimeter of a triangullar field is 144 m and the ratio of the sides is 3 : 4 : 5. Find the area of the field.
Solution:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
, where,
 
It is given the sides of a triangular field are in the ratio 3:4:5 and perimeter=144 m
Therefore, a: b: c = 3:4:5
We will assume the sides of triangular field as
 
Substituting the value of x in, we get sides of the triangle as
v
(v ) (v )
Page 3


        
     
                                    
            
               
                                    
                
 
 
                     
             
 
 
 
 
        
Question:26
Find the area of a triangle whose base and altitude are 5 cm and 4 cm respectively.
Solution:
Given, base = 5 cm; height = 4 cm
Area of the triangle = 
1
2
×Base ×Height
Question:27
Find the area of a triangle whose sides are 3 cm, 4 cm and 5 cm respectively.
Solution:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
, where
Therefore the area of a triangle, say having sides 3 cm, 4 cm and 5 cm is given by
a = 3 cm ; b = 4 cm ; c = 5 cm
Now, area 
= v 6(6 -3)(6 -4)(6 -5) = v 6 ×3 ×2 ×1 = v 36 = 6 cm
2
Question:28
Find the area of an isosceles triangle having the base x cm and one side y cm.
Solution:
Let us assume triangle ABC be the given isosceles triangle having sides AB = AC and base BC. The area of a triangle ABC, say A having given sides AB and AC equals to y cm and
given base BC equals to x cm is given by
Where,
Base = BC = x cm; Height = 
y
2
-
x
2
4
A =
1
2
(Base ×Height) =
1
2
×x y
2
-
x
2
4
=
x
2
y
2
-
x
2
4
Question:29
Find the area of an equilateral triangle having each side 4 cm.
Solution:
Area of an equilateral triangle having each side a cm is given by 
Area of the given equilateral triangle having each equal side equal to 4 cm is given by
a = 4 cm
Question:30
Find the area of an equilateral triangle having each side x cm.
Solution:
Area of an equilateral triangle, say A having each side a cm is given by 
Area of the given equilateral triangle having each equal side equal to x cm is given by
a = x cm
Question:31
The perimeter of a triangullar field is 144 m and the ratio of the sides is 3 : 4 : 5. Find the area of the field.
Solution:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
, where,
 
It is given the sides of a triangular field are in the ratio 3:4:5 and perimeter=144 m
Therefore, a: b: c = 3:4:5
We will assume the sides of triangular field as
 
Substituting the value of x in, we get sides of the triangle as
v
(v ) (v )
Area of a triangular field, say A having sides a, b , c and s as semi-perimeter is given by
Question:32
Find the area of an equilateral triangle having altitude h cm.
Solution:
Altitude of a equilateral triangle, having side a is given by
Substituting the given value of altitude h cm, we get
 
Area of a equilateral triangle, say A having each side a cm is given by 
Area of the given equilateral triangle having each equal side equal to  is given by; 
Question:33
Let ? be the area of a triangle. Find the area of a triangle whose each side is twice the side of the given triangle.
Solution:
We are given assumed value  is the area of a given triangle ABC
We assume the sides of the given triangle ABC be a, b, c
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
Where,
We take the sides of a new triangle as 2a, 2b, 2c that is twice the sides of previous one
Now, the area of a triangle having sides 2a, 2b, and 2c and  as semi-perimeter is given by,
, where
Now,
Question:34
If each side of a triangle is doubled, the find percentage increase in its area.
Solution:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
Page 4


        
     
                                    
            
               
                                    
                
 
 
                     
             
 
 
 
 
        
Question:26
Find the area of a triangle whose base and altitude are 5 cm and 4 cm respectively.
Solution:
Given, base = 5 cm; height = 4 cm
Area of the triangle = 
1
2
×Base ×Height
Question:27
Find the area of a triangle whose sides are 3 cm, 4 cm and 5 cm respectively.
Solution:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
, where
Therefore the area of a triangle, say having sides 3 cm, 4 cm and 5 cm is given by
a = 3 cm ; b = 4 cm ; c = 5 cm
Now, area 
= v 6(6 -3)(6 -4)(6 -5) = v 6 ×3 ×2 ×1 = v 36 = 6 cm
2
Question:28
Find the area of an isosceles triangle having the base x cm and one side y cm.
Solution:
Let us assume triangle ABC be the given isosceles triangle having sides AB = AC and base BC. The area of a triangle ABC, say A having given sides AB and AC equals to y cm and
given base BC equals to x cm is given by
Where,
Base = BC = x cm; Height = 
y
2
-
x
2
4
A =
1
2
(Base ×Height) =
1
2
×x y
2
-
x
2
4
=
x
2
y
2
-
x
2
4
Question:29
Find the area of an equilateral triangle having each side 4 cm.
Solution:
Area of an equilateral triangle having each side a cm is given by 
Area of the given equilateral triangle having each equal side equal to 4 cm is given by
a = 4 cm
Question:30
Find the area of an equilateral triangle having each side x cm.
Solution:
Area of an equilateral triangle, say A having each side a cm is given by 
Area of the given equilateral triangle having each equal side equal to x cm is given by
a = x cm
Question:31
The perimeter of a triangullar field is 144 m and the ratio of the sides is 3 : 4 : 5. Find the area of the field.
Solution:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
, where,
 
It is given the sides of a triangular field are in the ratio 3:4:5 and perimeter=144 m
Therefore, a: b: c = 3:4:5
We will assume the sides of triangular field as
 
Substituting the value of x in, we get sides of the triangle as
v
(v ) (v )
Area of a triangular field, say A having sides a, b , c and s as semi-perimeter is given by
Question:32
Find the area of an equilateral triangle having altitude h cm.
Solution:
Altitude of a equilateral triangle, having side a is given by
Substituting the given value of altitude h cm, we get
 
Area of a equilateral triangle, say A having each side a cm is given by 
Area of the given equilateral triangle having each equal side equal to  is given by; 
Question:33
Let ? be the area of a triangle. Find the area of a triangle whose each side is twice the side of the given triangle.
Solution:
We are given assumed value  is the area of a given triangle ABC
We assume the sides of the given triangle ABC be a, b, c
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
Where,
We take the sides of a new triangle as 2a, 2b, 2c that is twice the sides of previous one
Now, the area of a triangle having sides 2a, 2b, and 2c and  as semi-perimeter is given by,
, where
Now,
Question:34
If each side of a triangle is doubled, the find percentage increase in its area.
Solution:
The area of a triangle having sides a, b, c and s as semi-perimeter is given by,
  
Where,
We take the sides of a new triangle as 2a, 2b, 2c that is twice the sides of previous one
Now, the area of a triangle having sides 2a, 2b, and 2c and  as semi-perimeter is given by,
Where,
Now,
Therefore, increase in the area of the triangle
Percentage increase in area 
Question:35
If each side of a equilateral triangle is tripled then what is the percentage increase in the area of the triangle?
Solution:
Area of an equilateral triangle having each side a cm is given by 
Now, Area of an equilateral triangle, say  if each side is tripled is given by
a = 3a
Therefore, increase in area of triangle
Percentage increase in area