KC Sinha Mathematics Solution Class 10 Chapter 5 Triangles Exercise 5.1

Exercise 5.1
Exercise 5.2
Exercise 5.3
Exercise 5.4
Exercise 5.5

Exercise 5.1

Question 1

Fill in the blanks with the correct word given in brackets:
(i) All squares are having the same length of sides are……….
[similar, congruent, both congruent and similar]

(ii) All circles having the same radius are ……….
[similar, congruent, both congruent and similar]

(iii) All rhombuses having one angle 90^o ………. [similar, congruent]

(iv) All photographs of a given building made by the same negative are ………..
[similar, congruent, both congruent and similar]

(v) Two polygons having equal numbers of sides are similar if their corresponding angles are equal and their corresponding sides are ……….
[equal, proportional]

Sol :
(i)

Both congruent and similar because
(a) all squares have same shape and size
(b) their corresponding angles are equal
(c) their corresponding sides are proportional
(ii)

Both congruent and similar because all circles have same shape and size
(iii) Similar because all rhombuses have the same angle, but size can vary.
(iv) Similar because all photographs have the same shape but not necessarily the same size.
(v) Two polygons having equal numbers of sides are similar if their corresponding angles are equal and their corresponding sides are Proportional

Question 2

State which of the following statements are true and which are false:

(i) Two similar figures are congruent.
(ii) All congruent figures are similar.
(iii) All isosceles triangles are similar.
(iv) All right-angled triangles are similar.
(v) All squares are similar.
(vi) All rectangles are similar.
(vii) Two photographs of a person made by the same negative are similar.
(viii) Two photographs of a person one at the age of 5 years and other at the age of 50 years are similar.
Sol :
(i) This statement is false because all the congruent figures are similar, but similar figures need not be congruent.
E.g. Two equilateral triangles having sides 1cm and 2cm.

In case of equilateral triangles, all the sides are equal, and all the angles are of 60°.
But here, their corresponding angles are equal, but sides of triangle ABC and PQR are not equal in length.
So, they are similar figures but not congruent.
(ii) This statement is true because all congruent figures are similar, but similar figures need not be congruent.
(iii) This statement is false because for two triangles to be similar to the angles in one triangle must have the same values as the angles in the other triangle. The sides must be proportionate.
E.g.

These are the two isosceles triangles having two equal sides, but we can see that the sides are not proportionate.
(iv) This statement is false.
Suppose these are two right-angled triangles

Here, both of the triangles are right-angled, but other corresponding two angles are not equal. So, these are not similar figures.
(v) This statement is true because all the angles in a square are right angles and all the sides are equal. Hence, a smaller square can be enlarged to the size of a larger square, and vice-versa is also true.
(vi) This statement is false because similarity preserves the ratio of length. Therefore, two rectangles with a different ratio between their sides cannot be similar.
(vii) This statement is true because photographs are produced by projecting the image from a negative through an enlarger to a photographic paper. The enlarger reproduces the image from the negative but makes it bigger. The images are not identical and are not of the same size, but they are similar.

(viii) This statement is false because here the photograph of a person is taken at the different ages.

Question 3

Give two examples of:

(i) Congruent figures.
(ii) Similar figures which are not congruent.
(iii) Non-similar figures.
Sol :
(i) (a) Two circles having radii 2cm and different centres

In this, both of them have the same radii, but their centres are different.
(b) Two squares having the same length of side 5cm

We know that in a square all the sides are equal and all angles are of 90°. So, these two squares are congruent.
(ii) (a) Two equilateral triangles having sides 1cm and 2cm.

Both of the figures are of circle but they are having different radii. So, they are similar but not congruent.

(iii) (a) A rhombus and a rectangle
In the case of a rhombus, all the sides are equal, and the angles can either be right angles or combination of acute and obtuse angles but in rectangle all angles are equal, and opposite sides are equal.
Hence, a rhombus and a rectangle are non-similar figures.
(b)

Here, both of the triangles are right-angled but other two angles are not equal. So, these are not similar figures.

Question 4

State whether the following right-angled triangles are similar or not:

Sol :
Two polygons of a same number of sides are similar, if
a) all the corresponding angles are equal.
b) all the corresponding sides are in the same ratio (or proportion)
In case of right-angled triangle PQR and ABC
$\frac{P Q}{A B}=\frac{8}{4}=2$
$\frac{\mathrm{PR}}{\mathrm{AC}}=\frac{10}{5}=2$
and $\frac{Q R}{B C}=\frac{6}{3}=2$
The corresponding sides of a right-angled triangle ABC and PQR are proportional, and their corresponding angles are not equal. Hence, triangles ABC and PQR are not similar.

Question 5

State whether the following rectangles are similar or not.

Sol :
Two polygons of the same number of sides are similar, if
a) All the corresponding angles are equal.
b) all the corresponding sides are in the same ratio (or proportion)
In case of rectangles ABCD and PQRS
$\frac{A D}{P S}=\frac{5}{2}$
$\frac{A B}{P Q}=\frac{10}{4}=\frac{5}{2}$
And it is given that both are rectangles and we know that, in rectangle all angles are of 90°
The corresponding sides of a rectangle ABCD and PQRS are proportional, and their corresponding angles are equal. Hence, rectangles ABCD and PQRS are similar.

Question 6

State whether the following quadrilaterals are similar or not:

$\frac{A D}{P S}=\frac{5}{2}$

$\frac{A B}{P Q}=\frac{5}{4}$

$\frac{B C}{Q R}=\frac{5}{2}$

$\frac{C D}{R S}=\frac{5}{4}$

and ∠A=∠B =∠C =∠D =∠P =∠Q =∠R=∠S =90°
The corresponding sides of a quadrilateral ABCD and PQRS are not proportional. Hence, quadrilaterals ABCD and PQRS are not similar.

Question 7 A

State whether the following pair of polygons are similar or not.

Sol :
Two polygons of a same number of sides are similar, if
a) all the corresponding angles are equal.
b) all the corresponding sides are in the same ratio (or proportion)
In case of polygons ABCD and PQRS

$\frac{A D}{P S}=\frac{2}{2}=1$

$\frac{A B}{P Q}=\frac{4}{4}=1$

$\frac{B C}{Q R}=\frac{2}{2}=1$

$\frac{C D}{R S}=\frac{4}{4}=1$

and ∠A=∠B =∠C =∠D =90° but ∠P ,∠Q ,∠R,∠S≠90°
The corresponding sides of a polygon ABCD and PQRS are proportional, but their corresponding angles are not equal. Hence, polygon ABCD and PQRS are not similar.

Question 7 B

State whether the following pair of polygons are similar or not.

Sol :
Two polygons of a same number of sides are similar if
a) all the corresponding angles are equal.
b) all the corresponding sides are in the same ratio (or proportion)
In case of polygons ABCD and PQRS

$\frac{A D}{P S}=\frac{2.1}{4.2}=\frac{1}{2}$

$\frac{A B}{P Q}=\frac{1.5}{3.0}=\frac{1}{2}$

$\frac{B C}{Q R}=\frac{2.5}{5.0}=\frac{1}{2}$

$\frac{C D}{R S}=\frac{2.4}{4.8}=\frac{1}{2}$

and ∠A=∠P =105°, ∠B =∠Q =100°, ∠C =∠R=70°, ∠D=∠S =85°
The corresponding sides of a polygon ABCD and PQRS are proportional, and their corresponding angles are also equal. Hence, polygon ABCD and PQRS are similar.

Question 7 C

State whether the following pair of polygons are similar or not.

$\frac{A D}{P S}=\frac{3}{3}=1$

$\frac{A B}{P Q}=\frac{3}{3.5}$

$\frac{B C}{Q R}=\frac{3}{3}=1$

$\frac{C D}{R S}=\frac{3}{3.5}$

Clearly, the corresponding sides of a polygon ABCD and PQRS are not proportional. Hence, polygon ABCD and PQRS are not similar.

Question 7 D

State whether the following pair of polygons are similar or not.

Sol :
Two polygons of the same number of sides are similar if
a) all the corresponding angles are equal.
b) all the corresponding sides are in the same ratio (or proportion)
In case of polygons □ ABCD and ◊ ABCD

$\frac{A B}{A B}=\frac{2.1}{4.2}=2$

$\frac{B C}{B C}=\frac{2.1}{4.2}=2$

$\frac{C D}{C D}=\frac{2.1}{4.2}=2$

$\frac{D A}{D A}=\frac{2.1}{4.2}=2$

The corresponding sides of a polygon ABCD and ABCD are proportional, but their corresponding angles are not equal as we can see the first figure is of a square (all angles are of 90°) and other is of a rhombus (in rhombus the diagonal meet in the middle at a right angle). Hence, polygon ABCD and ABCD are not similar.

S.no	Chapters	Links
1	Real numbers	Exercise 1.1 Exercise 1.2 Exercise 1.3 Exercise 1.4
2	Polynomials	Exercise 2.1 Exercise 2.2 Exercise 2.3
3	Pairs of Linear Equations in Two Variables	Exercise 3.1 Exercise 3.2 Exercise 3.3 Exercise 3.4 Exercise 3.5
4	Trigonometric Ratios and Identities	Exercise 4.1 Exercise 4.2 Exercise 4.3 Exercise 4.4
5	Triangles	Exercise 5.1 Exercise 5.2 Exercise 5.3 Exercise 5.4 Exercise 5.5
6	Statistics	Exercise 6.1 Exercise 6.2 Exercise 6.3 Exercise 6.4
7	Quadratic Equations	Exercise 7.1 Exercise 7.2 Exercise 7.3 Exercise 7.4 Exercise 7.5
8	Arithmetic Progressions (AP)	Exercise 8.1 Exercise 8.2 Exercise 8.3 Exercise 8.4
9	Some Applications of Trigonometry: Height and Distances	Exercise 9.1
10	Coordinates Geometry	Exercise 10.1 Exercise 10.2 Exercise 10.3 Exercise 10.4
11	Circles	Exercise 11.1 Exercise 11.2
12	Constructions	Exercise 12.1
13	Area related to Circles	Exercise 13.1
14	Surface Area and Volumes	Exercise 14.1 Exercise 14.2 Exercise 14.3 Exercise 14.4
15	Probability	Exercise 15.1