NCERT Solutions for Class 7 Maths

NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.5 are part of NCERT Solutions for Class 7 Maths. Here we have given NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.5.

• The Triangle and its Properties Class 7 Ex 6.1
• The Triangle and its Properties Class 7 Ex 6.2
• The Triangle and its Properties Class 7 Ex 6.3
• The Triangle and its Properties Class 7 Ex 6.4
• The Triangle and its Properties Class 7 MCQ

NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.5

Question 1.
PQR is a triangle right-angled at P. If PQ = 10 cm and PR = 24 cm, find QR.

Solution:
QR² = 10² + 24² By Pythagoras Property
⇒ = 100 + 576 = 676
⇒ QR = 26 cm.

Question 2.
ABC is a triangle right-angled at C. If AB – 25 cm and AC = 7 cm, find BC.

Solution:
AC² + BC² = AB² By Pythagoras Property
⇒ 7² + BC² = 25²
⇒ 49 + BC² = 625
⇒ BC² = 625 – 49
⇒ BC² = 576
⇒ BC = 24 cm.

Question 3.
A 15 m long ladder reached a window 12 m high from the ground on placing it against a wall at a distance a. Find the distance of the foot of the ladder from the wall.

Solution:
Let the distance of the foot of the ladder from the wall be a m. Then,

Hence, the distance of the foot of the ladder from the wall is 9 m.

Question 4.
Which of the following can be the sides of a right triangle ?

1. 2.5 cm, 6.5 cm, 6 cm.
2. 2 cm, 2 cm, 5 cm.
3. 1.5 cm, 2 cm, 2.5 cm.

In the case of right-angled triangles, identify the right angles.
Solution:
1. 2.5 cm, 6.5 cm, 6 cm We see that
(2.5)² + 6² = 6.25 + 36 = 42.25 = (6.5)²
Therefore, the given lengths can be the sides of a right triangle. Also, the angle between the lengths, 2.5 cm and 6 cm is a right angle.

2. 2 cm, 2 cm, 5 cm
∵ 2 + 2 = 4 \(\ngtr\) 5
∴ The given lengths cannot be the sides of a triangle
The sum of the lengths of any two sides of a triangle is greater than the third side

3. 1.5 cm, 2 cm, 2.5 cm We find that
1.5² + 2² = 2.25 + 4 = 6.25 = 2.5²
Therefore, the given lengths can be the sides of a right triangle.
Also, the angle between the lengths 1.5 cm and 2 cm is a right angle.

Question 5.
A tree is broken at a height of 5 m from the ground and its top touches the ground at a distance of 12 m from the base of the tree. Find the original height of the tree.
Solution:
AC = CD Given
In right angled triangle DBC, DC² = BC² + BD²
by Pythagoras Property = 5² + 12² = 25 + 144 = 169

⇒ DC = 13 ⇒ AC = 13
⇒ AB = AC + BC = 13 + 5 = 18
Therefore, the original height of the tree = 18 m.

Question 6.
Angles Q and R of a ∆ PQR are 25° and 65°. Write which of the following is true:

(i) PQ² + QR² = RP²
(ii) PQ² + RP² = QR²
(iii) RP² + QR² = PQ²
Solution:
(ii) PQ² + RP² = QR² is true.

Question 7.
Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm.

Solution:
In right-angled triangle DAB, AB² + AD² = BD²
⇒ 40² + AD² = 41² ⇒ AD² = 41² – 40²
⇒ AD² = 1681 – 1600
⇒ AD² = 81 ⇒ AD = 9
∴ Perimeter of the rectangle = 2(AB + AD) = 2(40 + 9) = 2(49) = 98 cm
Hence, the perimeter of the rectangle is 98 cm.

Question 8.
The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter.

Solution:
Let ABCD be a rhombus whose diagonals BD and AC are of lengths 16 cm and 30 cm respectively.
Let the diagonals BD and AC intersect each other at O.
Since the diagonals of a rhombus bisect each other at right angles. Therefore
BO = OD = 8 cm,
AO = OC = 15 cm,
∠AOB = ∠BOC
= ∠COD = ∠DOA = 90°
In right-angled triangle AOB.
AB² = OA² + OB²
By Pythagoras Property
⇒ AB² = 15² + 8²
⇒ AB² = 225 + 64
⇒ AB² = 289
⇒ AB = 17cm
Therefore, perimeter of the rhombus ABCD = 4 side = 4 AB = 4 × 17 cm = 68 cm
Hence, the perimeter of the rhombus is 68 cm.

We hope the NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.5 help you. If you have any query regarding NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.5, drop a comment below and we will get back to you at the earliest.

NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.4 are part of NCERT Solutions for Class 7 Maths. Here we have given NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.4.

• The Triangle and its Properties Class 7 Ex 6.1
• The Triangle and its Properties Class 7 Ex 6.2
• The Triangle and its Properties Class 7 Ex 6.3
• The Triangle and its Properties Class 7 Ex 6.5
• The Triangle and its Properties Class 7 MCQ

NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.4

Question 1.
Is it possible to have a triangle with the following sides?

1. 2 cm, 3 cm, 5 cm
2. 3 cm, 6 cm, 7 cm
3. 6 cm, 3 cm, 2 cm.

Solution:

1. Since, 2 + 3 > 5
   So the given side lengths cannot form a triangle.
2. We have, 3 + 6 > 7, 3 + 7 > 6 and 6 + 7 > 3
   i. e., the sum of any two sides is greater than the third side.
   So, these side lengths form a triangle.
3. We have, 6 + 3 > 2, 3 + 2 \(\ngtr \) 6
   So, the given side lengths cannot form a triangle.

Question 2.
Take any point O in the interior of a triangle PQR. Is

1. OP + OQ > PQ ?
2. OQ + OR > QR?
3. OR + OP > RP ?

Solution:

1. Yes ! OP + OQ > PQ …(1)
   Sum of the lengths of any two sides of a triangle is greater than the length of the third side
2. Yes! OQ + OR > QR …(2)
   Sum of the lengths of any two sides of a triangle is greater than the length of the third side
3. Yes! OR + OP > RP …(3)
   Sum of the lengths of any two sides of a triangle is greater than the length of the third side

Question 3.
AM is a median of a triangle ABC. Is AB + BC + CA > 2 AM?
(Consider the sides of triangles ∆ ABM and ∆ AMC.)

Solution:
Using triangle inequality property in triangles ABM and AMC, we have
AB + BM > AM …(1) and, AC + MC > AM …(2)
Adding (1) and (2) on both sides, we get
AB + (BM + MC) + AC > AM + AM
⇒ AB + BC + AC > 2AM

Question 4.
ABCD is a quadrilateral.
Is AB + BC + CD + DA > AC + BD ?

Solution:
In ∆ ABC, AB + BC > AC …(1)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
In ∆ ACD, CD + DA > AC …(2)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
Adding (1) and (2),
AB + BC + CD + DA > 2AC …(3)
In ∆ ABD, AB + DA > BD …(4)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
In ∆ BCD, BC + CD > BD …(5)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
Adding (4) and (5),
AB + BC + CD + DA > 2BD …(6)
Adding (3) and (6),
2 [AB + BC + CD + DA] > 2 (AC + BD)
⇒ AB + BC + CD + DA > AC + BD.

Question 5.
ABCD is a quadrilateral. Is AB + BC + CD + DA < 2 (AC + BD)?

Solution:
In ∆ OAB, OA + OB > AB ….(1)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side.
In ∆ OBC, OB + OC > BC ….(2)
Sum of the lengths of any two sides of a triangle la greater than the length of the third side.
In ∆ OCA,OC + OA > CA ….(3)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
In ∆ OAD, OA + OD > AD ….(4)
Sum of the lengths of any two sides of a triangle is greater than the length of the third side
Adding (1), (2), (3) and (4),
2(OA + OB + OC + OD) > AB + BC + CD + DA
⇒ AB + BC + CD + DA < 2 (OA + OB + OC + OD)
⇒ AB + BC + CD + DA < 2(OA + OC + OB + OD)
⇒ AB + BC + CD + DA < 2 (AC + BD).

Question 6.
The lengths of the two sides of a triangle are 12 cm and 15 cm. Between what two measures should the length of the third side fall?
Solution:
Let x cm be the length of the third side.
∴ Sum of the lengths of any two sides of a triangle is greater than the length of the third side.
∴ We should have

∴ The length of the third side should be any length between 3 cm and 27 cm.

We hope the NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and it’s Properties Ex 6.4 help you. If you have any query regarding NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.4, drop a comment below and we will get back to you at the earliest.

NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.3 are part of NCERT Solutions for Class 7 Maths. Here we have given NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.3.

• The Triangle and its Properties Class 7 Ex 6.1
• The Triangle and its Properties Class 7 Ex 6.2
• The Triangle and its Properties Class 7 Ex 6.4
• The Triangle and its Properties Class 7 Ex 6.5
• The Triangle and its Properties Class 7 MCQ

NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.3

Question 1.
Find the value of the unknown x in the following diagrams:

Solution:

Question 2.
Find the values of the unknowns x and y in the following diagrams:

Solution:

We hope the NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.3 help you. If you have any query regarding NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.3, drop a comment below and we will get back to you at the earliest.

NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.2 are part of NCERT Solutions for Class 7 Maths. Here we have given NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.2.

• The Triangle and its Properties Class 7 Ex 6.1
• The Triangle and its Properties Class 7 Ex 6.3
• The Triangle and its Properties Class 7 Ex 6.4
• The Triangle and its Properties Class 7 Ex 6.5
• The Triangle and its Properties Class 7 MCQ

NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.2

Question 1.
Find the value of the unknown exterior angle x in the following diagrams:

Solution:

Question 2.
Find the value of the unknown interior angle x in the following figures:

Solution:

We hope the NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.2 help you. If you have any query regarding NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties Ex 6.2, drop a comment below and we will get back to you at the earliest.

NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles Ex 5.2 are part of NCERT Solutions for Class 7 Maths. Here we have given NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles Ex 5.2.

• Lines and Angles Class  7 Ex 5.1
• Lines and Angles Class  7 MCQ

NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles Ex 5.2

Question 1.
State the property that is used in each of the following statements?

1. If a || b, then ∠ 1 = ∠ 5.
2. If ∠ 4 = ∠ 6, then a || b.
3. If ∠ 4 + ∠ 5 = 180°, then a || b.

Solution:

1. Corresponding angle property.
2. Alternate interior angle property.
3. Interior angles on the same side of the transversal are supplementary.

Question 2.
In the following figure, identify:

1. the pairs of corresponding angles.
2. the pairs of alternate interior angles.
3. the pairs of interior angles on the same side of the transversal.
4. the vertically opposite angles.

Solution:

1. ∠1, ∠5; ∠2, ∠6; ∠3, ∠7 and ∠4, ∠8 are four pairs of corresponding angles.
2. ∠2, ∠8, and ∠3, ∠5 are two pairs of alternate interior angles.
3. ∠2, ∠5, and ∠3, ∠8 are two pairs of interior angles on the same side of the transversal.
4. ∠1, ∠3; ∠2, ∠4; ∠5, ∠7 and ∠6, ∠8 are four pairs of vertically opposite angles.

Question 3.
In the adjoining figure, p || q. Find the unknown angles.
Solution:
a = 55°, b = 125°, c = 55°, d = 125°, e = 55°, f = 55°.

Question 4.
Find the value of x in each of the following figures if l || m.

Solution:
(i) Since, l || m and t is a transversal.
∴ ∠x = (180° – 110°) = 70° [Corresponding angles, Linear pair]

(ii) if l || m and a is a transversal.
Then, ∠x = 1000 [Corresponding angles]

Question 5.
In the given figure, the arms of two angles are parallel. If ∠ ABC = 70°, then find

Solution:

1. 70°
2. 70°

Question 6.
In the given figures below, decide whether l is parallel to m.


Solution:

1. l is not parallel to m
2. l is not parallel to m
3. l || m
4. l is not parallel to m

We hope the NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles Ex 5.2 help you. If you have any query regarding NCERT Solutions for Class 7 Maths Chapter 5 Lines and Angles Ex 5.2, drop a comment below and we will get back to you at the earliest.