cbse ncert solutions for class 7 mathematics chapter 6
TRANSCRIPT
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 1 www.embibe.com
CBSE NCERT Solutions for Class 7 Mathematics Chapter 6 Back of Chapter Questions
EXERCISE 6.1
1. In βPQR, D is the mid-point of QRοΏ½οΏ½οΏ½οΏ½
PMοΏ½οΏ½οΏ½οΏ½ is ______
PD is _______
Is QM = MR?
Solution:
(i) Since, PM is perpendicular to QR. Hence, PMοΏ½οΏ½οΏ½οΏ½ is altitude.
(ii) Since, QD = DR. Hence, PD is median.
(iii) Since, QD = DR. Hence, QM β MR.
2. Draw rough sketches for the following:
(A) In ΞABC, BE is a median.
(B) In ΞPQR, PQ and PR are altitudes of the triangle.
(C) In ΞXYZ, YL is an altitude in the exterior of the triangle.
Solution:
(A)
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 2 www.embibe.com
(B) Since PQ and PR are altitude. Hence ΞPQR is a right-angle triangle.
(C)
It can be observed that for ΞXYZ, YL is an altitude drawn exterior to side XZ which
is extended up to point L.
3. Verify by drawing a diagram if the median and altitude of an isosceles triangle
can be same.
Solution:
Let us draw a line segment AD perpendicular to BC. It is an altitude for this
triangle. It can be observed that the length of BD and DC is also same. Therefore,
AD is also a median of this triangle.
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 3 www.embibe.com
EXERCISE 6.2
1. Find the value of the unknown exterior angle x in the following diagrams:
Solution:
(i) As per Exterior angle theorem,
Exterior angle = Sum of interior opposite angles
Hence, x = 50Β° + 70Β°
β x = 120Β°
(ii) As per Exterior angle theorem,
Exterior angle = Sum of interior opposite angles
Hence, x = 65Β° + 45Β°
β x = 110Β°
(iii) As per Exterior angle theorem,
Exterior angle = Sum of interior opposite angles
Hence, x = 40Β° + 30Β°
β x = 70Β°
(iv) As per Exterior angle theorem,
Exterior angle = Sum of interior opposite angles
Hence, x = 60Β° + 60Β°
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 4 www.embibe.com
β x = 120Β°
(v) As per Exterior angle theorem,
Exterior angle = Sum of interior opposite angles
Hence, x = 50Β° + 50Β°
β x = 100Β°
(vi) As per Exterior angle theorem,
Exterior angle = Sum of interior opposite angles
Hence, x = 30Β° + 60Β°
β x = 90Β°
2. Find the value of the unknown interior angle x in the following figures:
Solution:
(i) As per Exterior angle theorem,
Exterior angle = Sum of interior opposite angles
Hence, x + 50Β° = 115Β°
β x = 115Β° β 50Β° = 65Β°
(ii) As per Exterior angle theorem,
Exterior angle = Sum of interior opposite angles
Hence, 70Β° + x = 100Β°
β x = 100Β° β 70Β° = 30Β°
(iii) As per Exterior angle theorem,
Exterior angle = Sum of interior opposite angles
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 5 www.embibe.com
Hence, x + 90Β° = 125Β°
β x = 125Β° β 90Β° = 35Β°
(iv) As per Exterior angle theorem,
Exterior angle = Sum of interior opposite angles
Hence, x + 60Β° = 120Β°
β x = 120Β° β 60Β° = 60Β°
(v) As per Exterior angle theorem,
Exterior angle = Sum of interior opposite angles
Hence, x + 30Β° = 80Β°
β x = 80Β° β 30Β° = 50Β°
(vi) As per Exterior angle theorem,
Exterior angle = Sum of interior opposite angles
Hence, x + 35Β° = 75Β°
β x = 75o β 35o = 40o
EXERCISE 6.3
1. Find the value of the unknown x in the following diagrams:
Solution:
(i) As per angle sum property of a triangle,
The total measure of the three angles of a triangle is 180Β°.
Hence, x + 50Β° + 60Β° = 180Β°
β x + 110Β° = 180Β°
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 6 www.embibe.com
β x = 180Β° β 110Β° = 70Β°
(ii) As per angle sum property of a triangle,
The total measure of the three angles of a triangle is 180Β°.
Hence, x + 90Β° + 30Β° = 180Β°
β x + 120Β° = 180Β°
β x = 180Β° β 120Β° = 60Β°
(iii) As per angle sum property of a triangle,
The total measure of the three angles of a triangle is 180Β°.
Hence, x + 30Β° + 110Β° = 180Β°
β x + 140Β° = 180Β°
β x = 180Β° β 140Β° = 40Β°
(iv) As per angle sum property of a triangle,
The total measure of the three angles of a triangle is 180Β°.
Hence, 50Β° + x + x = 180Β°
β 50Β° + 2x = 180Β°
β 2x = 180Β° β 50Β° = 130Β°
β x = 130Β°
2= 65Β°
(v) As per angle sum property of a triangle,
The total measure of the three angles of a triangle is 180Β°.
Hence, x + x + x = 180Β°
β 3x = 180Β°
β x = 180Β°
3= 60Β°
(vi) As per angle sum property of a triangle,
The total measure of the three angles of a triangle is 180Β°.
Hence, x + 2x + 90Β° = 180Β°
β 3x = 180Β° β 90Β° = 90Β°
β x = 90Β°
3= 30Β°
2. Find the values of the unknowns x and y in the following diagrams:
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 7 www.embibe.com
Solution:
(i) y + 120Β° = 180Β° (Linear pair)
β y = 180Β° β 120Β° = 60Β°
Again, as per angle sum property of a triangle,
The total measure of the three angles of a triangle is 180Β°.
Hence, x + y + 50Β° = 180Β°
β x + 60Β° + 50Β° = 180Β°
β x + 110Β° = 180Β°
β x = 180Β° β 110Β° = 70Β°
Hence, the value of x and y are 70Β° and 60Β° respectively.
(ii) y = 80Β° (Vertically opposite angles)
Again, as per angle sum property of a triangle,
The total measure of the three angles of a triangle is 180Β°.
Hence, y + x + 50Β° = 180Β°
β 80Β° + x + 50Β° = 180Β°
β x + 130Β° = 180Β°
β x = 180Β° β 130Β° = 50Β°
Hence, the value of x and y are 50Β° and 80Β° respectively.
(iii) As per angle sum property of a triangle,
The total measure of the three angles of a triangle is 180Β°.
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 8 www.embibe.com
Hence, y + 50Β° + 60Β° = 180Β° (Angle sum property)
β y = 180Β° β 60Β° β 50Β° = 70Β°
x + y = 180Β° (Linear pair)
β x = 180Β° β y = 180Β° β 70Β° = 110Β°
Hence, the value of x and y are 110Β° and 70Β° respectively.
(iv) x = 60Β° (Vertically opposite angles)
Again, as per angle sum property of a triangle,
The total measure of the three angles of a triangle is 180Β°.
Hence, 30Β° + x + y = 180Β°
β 30Β° + 60Β° + y = 180Β°
β y = 180Β° β 30Β° β 60Β° = 90Β°
Hence, the value of x and y are 60Β° and 90Β° respectively.
(v) y = 90Β° (Vertically opposite angles)
Again, as per angle sum property of a triangle,
The total measure of the three angles of a triangle is 180Β°.
Hence, x + x + y = 180Β°
β 2x + y = 180Β°
β 2x + 90Β° = 180Β°
β 2x = 180Β° β 90Β° = 90Β°
β x =90Β°
2= 45Β°
Hence, the value of x and y are 45Β° and 90Β° respectively.
(vi)
y = x (Vertically opposite angles)
a = x (Vertically opposite angles)
b = x (Vertically opposite angles)
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 9 www.embibe.com
Again, as per angle sum property of a triangle,
The total measure of the three angles of a triangle is 180Β°.
Hence, a + b + y = 180Β°
β x + x + x = 180Β°
β 3x = 180Β°
β x =180Β°
3= 60Β°
Hence, the value of y = x = 60Β°
EXERCISE 6.4
1. Is it possible to have a triangle with the following sides?
(i) 2 cm, 3 cm, 5 cm
(ii) 3 cm, 6 cm, 7cm
(iii) 6 cm, 3 cm, 2 cm
Solution:
In a triangle, the sum of the lengths of any two sides of a triangle is greater than
the length of the third side.
(i) Given that, the sides of the triangle are 2 cm, 3 cm, 5 cm.
It can be observed that,
2 + 3 = 5 cm
However, 5 cm = 5 cm
Hence, this triangle is not possible.
(ii) Given that, the sides of the triangle are 3 cm, 6 cm, 7 cm.
Here, 3 + 6 = 9 cm > 7 cm
6 + 7 = 13 cm > 3 cm
3 + 7 = 10 cm > 6 cm
Hence, this triangle is possible.
(iii) Given that, the sides of the triangle are 6 cm, 3 cm, 2 cm.
Here, 6 + 3 = 9 cm > 2 cm
However, 3 + 2 = 5 cm < 6 cm
Hence, this triangle is not possible.
2. Take any point O in the interior of a triangle PQR. Is
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 10 www.embibe.com
(i) OP + OQ > PQ?
(ii) OQ + OR > QR?
(iii) OR + OP > RP?
Solution:
Given that, if O is a point in the interior of a given triangle, then three triangles
ΞOPQ, ΞOQR, and ΞORP can be constructed. In a triangle, the sum of the lengths
of either two sides is always greater than the third side.
(i)
Yes, as ΞOPQ is a triangle with sides OP, OQ, and PQ.
β΄ OP + OQ > PQ
(ii)
Yes, as ΞOQR is a triangle with sides OR, OQ, and QR.
β΄ OQ + OR > QR
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 11 www.embibe.com
(iii)
Yes, as ΞORP is a triangle with sides OR, OP, and PR.
β΄ OR + OP > RP
3. AM is a median of a triangle ABC.
Is AB + BC + CA > 2AM?
(Consider the sides of triangles ΞABM and ΞAMC.)
Solution:
In a triangle, the sum of the lengths of any two sides is greater than the length of
the third side
In ΞABM,
AB + BM > AM β¦β¦..(i)
Similarly, in ΞACM,
AC + CM > AM β¦β¦..(ii)
Adding equation (i) and (ii),
AB + BM + MC + AC > AM + AM
β AB + BC + AC > 2AM
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 12 www.embibe.com
Therefore, the given expression is true.
4. ABCD is a quadrilateral
Is AB + BC + CD + DA > AC + BD?
Solution:
In a triangle, the sum of the lengths of either two sides is always greater than the
third side.
Consider ΞABC,
AB + BC > CA β¦β¦β¦β¦ (i)
In ΞBCD,
BC + CD > DB β¦β¦β¦β¦(ii)
In ΞCDA,
CD + DA > AC β¦β¦β¦β¦ (iii)
In ΞDAB,
DA + AB > DB β¦β¦β¦β¦ (iv)
Adding equations (i), (ii), (iii), and (iv), we obtain
AB + BC + BC + CD + CD + DA + DA + AB > AC + BD + AC + BD
β 2AB + 2BC + 2CD + 2DA > 2AC + 2BD
β 2(AB + BC + CD + DA) > 2(AC + BD)
β (π΄π΄π΄π΄ + π΄π΄πΆπΆ + πΆπΆπΆπΆ + πΆπΆπ΄π΄) > (π΄π΄πΆπΆ + π΄π΄πΆπΆ)
Therefore, the given expression is true.
5. π΄π΄π΄π΄πΆπΆπΆπΆ is a quadrilateral.
Is π΄π΄π΄π΄ + π΄π΄πΆπΆ + πΆπΆπΆπΆ + πΆπΆπ΄π΄ < 2(π΄π΄πΆπΆ + π΄π΄πΆπΆ)?
Solution:
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 13 www.embibe.com
In a triangle, the sum of the lengths of either two sides is always greater than the
third side.
Consider π₯π₯π₯π₯π΄π΄π΄π΄,
π₯π₯π΄π΄ + π₯π₯π΄π΄ > π΄π΄π΄π΄ β¦β¦β¦ (i)
In π₯π₯π₯π₯π΄π΄πΆπΆ,
π₯π₯π΄π΄ + π₯π₯πΆπΆ > π΄π΄πΆπΆ β¦β¦β¦ (ii)
In π₯π₯π₯π₯πΆπΆπΆπΆ,
π₯π₯πΆπΆ + π₯π₯πΆπΆ > πΆπΆπΆπΆ β¦β¦β¦β¦ (iii)
In π₯π₯π₯π₯πΆπΆπ΄π΄,
π₯π₯πΆπΆ + π₯π₯π΄π΄ > πΆπΆπ΄π΄ β¦β¦β¦ (iv)
Adding equations (ππ), (ππππ), (ππππππ), ππππππ (ππππ), we obtain
π₯π₯π΄π΄ + π₯π₯π΄π΄ + π₯π₯π΄π΄ + π₯π₯πΆπΆ + π₯π₯πΆπΆ + π₯π₯πΆπΆ + π₯π₯πΆπΆ + π₯π₯π΄π΄ > π΄π΄π΄π΄ + π΄π΄πΆπΆ + πΆπΆπΆπΆ + πΆπΆπ΄π΄
β 2π₯π₯π΄π΄ + 2π₯π₯π΄π΄ + 2π₯π₯πΆπΆ + 2π₯π₯πΆπΆ > π΄π΄π΄π΄ + π΄π΄πΆπΆ + πΆπΆπΆπΆ + πΆπΆπ΄π΄
β 2π₯π₯π΄π΄ + 2π₯π₯πΆπΆ + 2π₯π₯π΄π΄ + 2π₯π₯πΆπΆ > π΄π΄π΄π΄ + π΄π΄πΆπΆ + πΆπΆπΆπΆ + πΆπΆπ΄π΄
2(π₯π₯π΄π΄ + π₯π₯πΆπΆ) + 2(π₯π₯π΄π΄ + π₯π₯πΆπΆ) > π΄π΄π΄π΄ + π΄π΄πΆπΆ + πΆπΆπΆπΆ + πΆπΆπ΄π΄
β 2(π΄π΄πΆπΆ) + 2(π΄π΄πΆπΆ) > π΄π΄π΄π΄ + π΄π΄πΆπΆ + πΆπΆπΆπΆ + πΆπΆπ΄π΄
β 2(π΄π΄πΆπΆ + π΄π΄πΆπΆ) > π΄π΄π΄π΄ + π΄π΄πΆπΆ + πΆπΆπΆπΆ + πΆπΆπ΄π΄
β π΄π΄π΄π΄ + π΄π΄πΆπΆ + πΆπΆπΆπΆ + πΆπΆπ΄π΄ < 2(π΄π΄πΆπΆ + π΄π΄πΆπΆ)
Therefore, the given expression is true.
6. The lengths of two sides of a triangle are 12 ππππ and 15 ππππ.
Between what two measures should the length of the third side fall?
Solution:
In a triangle, the sum of the lengths of either two sides is always greater than the
third side and also, the difference of the lengths of either two sides is always
lesser than the third side.
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 14 www.embibe.com
Therefore, the third side will be lesser than the sum of these two
(ππ. ππ. ,12 + 15 = 27) and also, it will be greater than the difference of these two
(ππ. ππ. ,15 β 12 = 3).
Therefore, the third side ranges between 3 ππππ and 27 ππππ.
EXERCISE 6.5
1. ππππππ is a triangle right angled at ππ. If ππππ = 10 ππππ and ππππ = 24 ππππ, find ππππ.
Solution:
By applying Pythagoras theorem to π₯π₯ππππππ,
(ππππ)2 + (ππππ)2 = (ππππ)2
β (10)2 + (24)2 = (ππππ)2
β 100 + 576 = (ππππ)2
β 676 = (ππππ)2
β ππππ = β676 ππππ = 26 ππππ
Therefore, ππππ = 26 ππππ
2. ABC is a triangle right angled at πΆπΆ. If π΄π΄π΄π΄ = 25 ππππ and π΄π΄πΆπΆ = 7 ππππ, find π΄π΄πΆπΆ.
Solution:
By applying Pythagoras theorem to π₯π₯π΄π΄π΄π΄πΆπΆ,
(π΄π΄πΆπΆ)2 + (π΄π΄πΆπΆ)2 = (π΄π΄π΄π΄)2
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 15 www.embibe.com
β (π΄π΄πΆπΆ)2 = (π΄π΄π΄π΄)2 β (π΄π΄πΆπΆ)2
β (π΄π΄πΆπΆ)2 = (25)2 β (7)2
β (π΄π΄πΆπΆ)2 = 625 β 49 = 576
β π΄π΄πΆπΆ = β576 ππππ = 24 ππππ
Therefore, π΄π΄πΆπΆ = 24 ππππ
3. A 15 ππ long ladder reached a window 12 ππ high from the ground on placing it
against a wall at a distance ππ. Find the distance of the foot of the ladder from the
wall.
Solution:
By applying Pythagoras theorem,
152 = 122 + ππ2
β 225 = 144 + ππ2
β ππ2 = 225 β 144 = 81
β ππ = 9 ππ
Therefore, the distance of the foot of the ladder from the wall is 9 ππ.
4. Which of the following can be the sides of a right triangle?
(i) 2.5 ππππ, 6.5 ππππ, 6 ππππ
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 16 www.embibe.com
(ii) 2 ππππ, 2 ππππ, 5 ππππ
(iii) 1.5 ππππ, 2 ππππ, 2.5 ππππ
In the case of right-angled triangles, identify the right angles.
Solution:
(i) 2.5 ππππ, 6.5 ππππ, 6 ππππ
2.52 = 6.25
6.52 = 42.25
62 = 36
It can be observed that,
36 + 6.25 = 42.25
β 62 + 2.52 = 6.52
The square of the length of one side is the sum of the squares of the
lengths of the remaining two sides. Hence, these are the sides of a right-
angled triangle with the side measuring 6.5 ππππ as the hypotenuse.
(ii) 2 ππππ, 2 ππππ, 5 ππππ
22 = 4
22 = 4
52 = 25
Here, 22 + 22 β 52
The square of the length of one side is not equal to the sum of the squares
of the lengths of the remaining two sides. Hence, these sides are not of a
right-angled triangle.
(iii) 1.5 ππππ, 2 ππππ, 2.5 ππππ
1.52 = 2.25
22 = 4
2.52 = 6.25
Here,
2.25 + 4 = 6.25
1.52 + 22 = 2.52
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 17 www.embibe.com
The square of the length of one side is the sum of the squares of the
lengths of the remaining two sides. Hence, these are the sides of a right-
angled triangle with the side measuring 2.5 ππππ as the hypotenuse.
5. A tree is broken at a height of 5 ππ from the ground and its top touches the ground
at a distance of 12 ππ from the base of the tree. Find the original height of the tree.
Solution:
In the given figure, π΄π΄πΆπΆ represents the unbroken part of the tree. Point πΆπΆ represents
the point where the tree broke and πΆπΆπ΄π΄ represents the broken part of the tree.
Triangle π΄π΄π΄π΄πΆπΆ, thus formed, is right-angled at π΄π΄.
Applying Pythagoras theorem in π₯π₯π΄π΄π΄π΄πΆπΆ,
(π΄π΄π΄π΄)2 + (π΄π΄πΆπΆ)2 = (π΄π΄πΆπΆ)2
β (π΄π΄πΆπΆ)2 = 122 + 52
β π΄π΄πΆπΆ2 = 25 + 144 = 169
β π΄π΄πΆπΆ = 13 ππ
Thus, original height of the tree = π΄π΄πΆπΆ + πΆπΆπ΄π΄ = 13 ππ + 5 ππ = 18 ππ
6. Angles ππ and ππ of a π₯π₯ππππππ are 25Β° and 65Β°.
Write which of the following is true:
(i) (ππππ)2 + (ππππ)2 = (ππππ)2
(ii) (ππππ)2 + (ππππ)2 = (ππππ)2
(iii) (ππππ)2 + (ππππ)2 = (ππππ)2
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 18 www.embibe.com
Solution:
The sum of the measures of all interior angles of a triangle is 180ππ.
Hence, β ππππππ + β ππππππ + β ππππππ = 180Β°
β 25Β° + 65Β° + β ππππππ = 180Β°
β 90Β° + β ππππππ = 180Β°
β β ππππππ = 180Β° β 90Β° = 90Β°
Therefore, π₯π₯ ππππππ is right-angled at point ππ.
Hence, (ππππ)2 + (ππππ)2 = (ππππ)2
Hence, (ππππ)2 + (ππππ)2 = (ππππ)2
Thus, (ii) is true.
7. Find the perimeter of the rectangle whose length is 40 ππππ and a diagonal is
41 ππππ.
Solution:
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 19 www.embibe.com
In a rectangle, all interior angles are of 90Β° measure. Therefore, Pythagoras
theorem can be applied here.
412 = 402 + π₯π₯2
β 1681 = 1600 + π₯π₯2
β π₯π₯2 = 1681 β 1600 = 81
β π₯π₯ = 9 ππππ
Perimeter = 2(πΏπΏπππππΏπΏπΏπΏβ + π΄π΄π΅π΅πππππππΏπΏβ)
= 2(π₯π₯ + 40) = 2(9 + 40) = 98 ππππ
Therefore, the perimeter of the given rectangle is 98 ππππ.
8. The diagonals of a rhombus measure 16 ππππ and 30 ππππ. Find its perimeter.
Solution:
Let π΄π΄π΄π΄πΆπΆπΆπΆ be a rhombus (all sides are of equal length) and its diagonals, π΄π΄πΆπΆ and
π΄π΄πΆπΆ, are intersecting each other at point π₯π₯. Diagonals in a rhombus bisect each
other at 90Β°. It can be observed that
π΄π΄π₯π₯ = π΄π΄πΆπΆ2
= 162
= 8 ππππ
Class- VII-CBSE-Mathematics The Triangle And Its Properties
Practice more on The Triangle And Its Properties Page - 20 www.embibe.com
π΄π΄π₯π₯ = π΄π΄πΆπΆ2
= 302
= 15
By applying Pythagoras theorem in ΞAOB,
OA2 + OB2 = AB2cm
β 82 + 152 = AB2
β 64 + 225 = AB2
β 289 = AB2
β AB = 17
Therefore, the length of the side of rhombus is 17 cm.
Perimeter of rhombus = 4 Γ Side of the rhombus = 4 Γ 17 = 68 cm
Hence, the perimeter of the rhombus = 68 cm.
⧫ ⧫ ⧫