Class 7 Exam > Class 7 Notes > Mathematics (Maths) Class 7 > RD Sharma Solutions: Properties of Triangles (Exercise 15.5)
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Page 1
1. State Pythagoras theorem and its converse.
Solution:
The Pythagoras Theorem:
In a right triangle, the square of the hypotenuse is always equal to the sum of the
squares of the other two sides.
Converse of the Pythagoras Theorem:
If the square of one side of a triangle is equal to the sum of the squares of the other two
sides, then the triangle is a right triangle, with the angle opposite to the first side as
right angle.
2. In right ?ABC, the lengths of the legs are given. Find the length of the hypotenuse
(i) a = 6 cm, b = 8 cm
(ii) a = 8 cm, b = 15 cm
(iii) a = 3 cm, b = 4 cm
(iv) a = 2 cm, b =1.5 cm
Solution:
(i) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 6
2
+ 8
2
c
2
= 36 + 64 = 100
c = 10 cm
(ii) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 8
2
+ 15
2
c
2
= 64 + 225 = 289
Page 2
1. State Pythagoras theorem and its converse.
Solution:
The Pythagoras Theorem:
In a right triangle, the square of the hypotenuse is always equal to the sum of the
squares of the other two sides.
Converse of the Pythagoras Theorem:
If the square of one side of a triangle is equal to the sum of the squares of the other two
sides, then the triangle is a right triangle, with the angle opposite to the first side as
right angle.
2. In right ?ABC, the lengths of the legs are given. Find the length of the hypotenuse
(i) a = 6 cm, b = 8 cm
(ii) a = 8 cm, b = 15 cm
(iii) a = 3 cm, b = 4 cm
(iv) a = 2 cm, b =1.5 cm
Solution:
(i) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 6
2
+ 8
2
c
2
= 36 + 64 = 100
c = 10 cm
(ii) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 8
2
+ 15
2
c
2
= 64 + 225 = 289
c = 17cm
(iii) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 3
2
+ 4
2
c
2
= 9 + 16 = 25
c = 5 cm
(iv) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 2
2
+ 1.5
2
c
2
= 4 + 2.25 = 6.25
c = 2.5 cm
3. The hypotenuse of a triangle is 2.5 cm. If one of the sides is 1.5 cm. find the length
of the other side.
Solution:
Let c be hypotenuse and the other two sides be b and a
According to the Pythagoras theorem, we have
c
2
= a
2
+ b
2
2.5
2
= 1.5
2
+ b
2
b
2
= 6.25 -2.25 = 4
b = 2 cm
Hence, the length of the other side is 2 cm.
4. A ladder 3.7 m long is placed against a wall in such a way that the foot of the ladder
is 1.2 m away from the wall. Find the height of the wall to which the ladder reaches.
Solution:
Page 3
1. State Pythagoras theorem and its converse.
Solution:
The Pythagoras Theorem:
In a right triangle, the square of the hypotenuse is always equal to the sum of the
squares of the other two sides.
Converse of the Pythagoras Theorem:
If the square of one side of a triangle is equal to the sum of the squares of the other two
sides, then the triangle is a right triangle, with the angle opposite to the first side as
right angle.
2. In right ?ABC, the lengths of the legs are given. Find the length of the hypotenuse
(i) a = 6 cm, b = 8 cm
(ii) a = 8 cm, b = 15 cm
(iii) a = 3 cm, b = 4 cm
(iv) a = 2 cm, b =1.5 cm
Solution:
(i) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 6
2
+ 8
2
c
2
= 36 + 64 = 100
c = 10 cm
(ii) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 8
2
+ 15
2
c
2
= 64 + 225 = 289
c = 17cm
(iii) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 3
2
+ 4
2
c
2
= 9 + 16 = 25
c = 5 cm
(iv) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 2
2
+ 1.5
2
c
2
= 4 + 2.25 = 6.25
c = 2.5 cm
3. The hypotenuse of a triangle is 2.5 cm. If one of the sides is 1.5 cm. find the length
of the other side.
Solution:
Let c be hypotenuse and the other two sides be b and a
According to the Pythagoras theorem, we have
c
2
= a
2
+ b
2
2.5
2
= 1.5
2
+ b
2
b
2
= 6.25 -2.25 = 4
b = 2 cm
Hence, the length of the other side is 2 cm.
4. A ladder 3.7 m long is placed against a wall in such a way that the foot of the ladder
is 1.2 m away from the wall. Find the height of the wall to which the ladder reaches.
Solution:
Let the height of the ladder reaches to the wall be h.
According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
3.7
2
= 1.2
2
+ h
2
h
2
= 13.69 – 1.44 = 12.25
h = 3.5 m
Hence, the height of the wall is 3.5 m.
5. If the sides of a triangle are 3 cm, 4 cm and 6 cm long, determine whether the
triangle is right-angled triangle.
Solution:
In the given triangle, the largest side is 6 cm.
We know that in a right angled triangle, the sum of the squares of the smaller sides
should be equal to the square of the largest side.
Therefore,
3
2
+ 4
2
= 9 + 16 = 25
But, 6
2
= 36
3
2
+ 4
2
= 25 which is not equal to 6
2
Hence, the given triangle is not a right angled triangle.
6. The sides of certain triangles are given below. Determine which of them are right
triangles.
(i) a = 7 cm, b = 24 cm and c= 25 cm
(ii) a = 9 cm, b = 16 cm and c = 18 cm
Solution:
(i) We know that in a right angled triangle, the square of the largest side is equal to the
Page 4
1. State Pythagoras theorem and its converse.
Solution:
The Pythagoras Theorem:
In a right triangle, the square of the hypotenuse is always equal to the sum of the
squares of the other two sides.
Converse of the Pythagoras Theorem:
If the square of one side of a triangle is equal to the sum of the squares of the other two
sides, then the triangle is a right triangle, with the angle opposite to the first side as
right angle.
2. In right ?ABC, the lengths of the legs are given. Find the length of the hypotenuse
(i) a = 6 cm, b = 8 cm
(ii) a = 8 cm, b = 15 cm
(iii) a = 3 cm, b = 4 cm
(iv) a = 2 cm, b =1.5 cm
Solution:
(i) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 6
2
+ 8
2
c
2
= 36 + 64 = 100
c = 10 cm
(ii) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 8
2
+ 15
2
c
2
= 64 + 225 = 289
c = 17cm
(iii) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 3
2
+ 4
2
c
2
= 9 + 16 = 25
c = 5 cm
(iv) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 2
2
+ 1.5
2
c
2
= 4 + 2.25 = 6.25
c = 2.5 cm
3. The hypotenuse of a triangle is 2.5 cm. If one of the sides is 1.5 cm. find the length
of the other side.
Solution:
Let c be hypotenuse and the other two sides be b and a
According to the Pythagoras theorem, we have
c
2
= a
2
+ b
2
2.5
2
= 1.5
2
+ b
2
b
2
= 6.25 -2.25 = 4
b = 2 cm
Hence, the length of the other side is 2 cm.
4. A ladder 3.7 m long is placed against a wall in such a way that the foot of the ladder
is 1.2 m away from the wall. Find the height of the wall to which the ladder reaches.
Solution:
Let the height of the ladder reaches to the wall be h.
According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
3.7
2
= 1.2
2
+ h
2
h
2
= 13.69 – 1.44 = 12.25
h = 3.5 m
Hence, the height of the wall is 3.5 m.
5. If the sides of a triangle are 3 cm, 4 cm and 6 cm long, determine whether the
triangle is right-angled triangle.
Solution:
In the given triangle, the largest side is 6 cm.
We know that in a right angled triangle, the sum of the squares of the smaller sides
should be equal to the square of the largest side.
Therefore,
3
2
+ 4
2
= 9 + 16 = 25
But, 6
2
= 36
3
2
+ 4
2
= 25 which is not equal to 6
2
Hence, the given triangle is not a right angled triangle.
6. The sides of certain triangles are given below. Determine which of them are right
triangles.
(i) a = 7 cm, b = 24 cm and c= 25 cm
(ii) a = 9 cm, b = 16 cm and c = 18 cm
Solution:
(i) We know that in a right angled triangle, the square of the largest side is equal to the
sum of the squares of the smaller sides.
Here, the larger side is c, which is 25 cm.
c
2
= 625
Given that,
a
2
+ b
2
= 7
2
+ 24
2
= 49 + 576
= 625
= c
2
Thus, the given triangle is a right triangle.
(ii) We know that in a right angled triangle, the square of the largest side is equal to the
sum of the squares of the smaller sides.
Here, the larger side is c, which is 18 cm.
c
2
= 324
Given that
a
2
+ b
2
= 9
2
+ 16
2
= 81 + 256
= 337 which is not equal to c
2
Thus, the given triangle is not a right triangle.
7. Two poles of heights 6 m and 11 m stand on a plane ground. If the distance
between their feet is 12 m. Find the distance between their tops.
(Hint: Find the hypotenuse of a right triangle having the sides (11 – 6) m = 5 m and 12
m)
Solution:
Let the distance between the tops of the poles is the distance between points A and B.
We can see from the given figure that points A, B and C form a right triangle, with AB as
the hypotenuse.
Page 5
1. State Pythagoras theorem and its converse.
Solution:
The Pythagoras Theorem:
In a right triangle, the square of the hypotenuse is always equal to the sum of the
squares of the other two sides.
Converse of the Pythagoras Theorem:
If the square of one side of a triangle is equal to the sum of the squares of the other two
sides, then the triangle is a right triangle, with the angle opposite to the first side as
right angle.
2. In right ?ABC, the lengths of the legs are given. Find the length of the hypotenuse
(i) a = 6 cm, b = 8 cm
(ii) a = 8 cm, b = 15 cm
(iii) a = 3 cm, b = 4 cm
(iv) a = 2 cm, b =1.5 cm
Solution:
(i) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 6
2
+ 8
2
c
2
= 36 + 64 = 100
c = 10 cm
(ii) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 8
2
+ 15
2
c
2
= 64 + 225 = 289
c = 17cm
(iii) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 3
2
+ 4
2
c
2
= 9 + 16 = 25
c = 5 cm
(iv) According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
Let c be hypotenuse and a and b be other two legs of right angled triangle
Then we have
c
2
= a
2
+ b
2
c
2
= 2
2
+ 1.5
2
c
2
= 4 + 2.25 = 6.25
c = 2.5 cm
3. The hypotenuse of a triangle is 2.5 cm. If one of the sides is 1.5 cm. find the length
of the other side.
Solution:
Let c be hypotenuse and the other two sides be b and a
According to the Pythagoras theorem, we have
c
2
= a
2
+ b
2
2.5
2
= 1.5
2
+ b
2
b
2
= 6.25 -2.25 = 4
b = 2 cm
Hence, the length of the other side is 2 cm.
4. A ladder 3.7 m long is placed against a wall in such a way that the foot of the ladder
is 1.2 m away from the wall. Find the height of the wall to which the ladder reaches.
Solution:
Let the height of the ladder reaches to the wall be h.
According to the Pythagoras theorem, we have
(Hypotenuse)
2
= (Base)
2
+ (Height)
2
3.7
2
= 1.2
2
+ h
2
h
2
= 13.69 – 1.44 = 12.25
h = 3.5 m
Hence, the height of the wall is 3.5 m.
5. If the sides of a triangle are 3 cm, 4 cm and 6 cm long, determine whether the
triangle is right-angled triangle.
Solution:
In the given triangle, the largest side is 6 cm.
We know that in a right angled triangle, the sum of the squares of the smaller sides
should be equal to the square of the largest side.
Therefore,
3
2
+ 4
2
= 9 + 16 = 25
But, 6
2
= 36
3
2
+ 4
2
= 25 which is not equal to 6
2
Hence, the given triangle is not a right angled triangle.
6. The sides of certain triangles are given below. Determine which of them are right
triangles.
(i) a = 7 cm, b = 24 cm and c= 25 cm
(ii) a = 9 cm, b = 16 cm and c = 18 cm
Solution:
(i) We know that in a right angled triangle, the square of the largest side is equal to the
sum of the squares of the smaller sides.
Here, the larger side is c, which is 25 cm.
c
2
= 625
Given that,
a
2
+ b
2
= 7
2
+ 24
2
= 49 + 576
= 625
= c
2
Thus, the given triangle is a right triangle.
(ii) We know that in a right angled triangle, the square of the largest side is equal to the
sum of the squares of the smaller sides.
Here, the larger side is c, which is 18 cm.
c
2
= 324
Given that
a
2
+ b
2
= 9
2
+ 16
2
= 81 + 256
= 337 which is not equal to c
2
Thus, the given triangle is not a right triangle.
7. Two poles of heights 6 m and 11 m stand on a plane ground. If the distance
between their feet is 12 m. Find the distance between their tops.
(Hint: Find the hypotenuse of a right triangle having the sides (11 – 6) m = 5 m and 12
m)
Solution:
Let the distance between the tops of the poles is the distance between points A and B.
We can see from the given figure that points A, B and C form a right triangle, with AB as
the hypotenuse.
By using the Pythagoras Theorem in ?ABC, we get
(11-6)
2
+ 12
2
= AB
2
AB
2
= 25 + 144
AB
2
= 169
AB = 13
Hence, the distance between the tops of the poles is 13 m.
8. A man goes 15 m due west and then 8 m due north. How far is he from the starting
point?
Solution:
Given a man goes 15 m due west and then 8 m due north
Let O be the starting point and P be the final point.
Then OP becomes the hypotenuse in the triangle.
So by using the Pythagoras theorem, we can find the distance OP.
OP
2
= 15
2
+ 8
2
OP
2
= 225 + 64
OP
2
= 289
OP = 17
Hence, the required distance is 17 m.
9. The foot of a ladder is 6 m away from a wall and its top reaches a window 8 m
above the ground. If the ladder is shifted in such a way that its foot is 8 m away from
the wall, to what height does its top reach?
Solution:
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