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Page 1
Circle
Practice Set 3.1
Q. 1. In the adjoining figure the radius of a circle with centre C is 6 cm, line AB is
a tangent at A. Answer the following questions.
(1) What is the measure of ?CAB? Why?
(2) What is the distance of point C from line AB? Why?
(3) d(A,B) = 6 cm, find d(B,C).
(4) What is the measure of ?ABC? Why?
Answer : (1) ere CA is the radius of the circle and A is the point of contact of the
tangent AB.
? ?CAB = 90° Using tangent-radius theorem which states that a tangent at any point of
a circle is perpendicular to the radius at the point of contact.
(2) CA is the radius of the circle which is perpendicular to the tangent AB.
So, the perpendicular distance of line AB from C = CA = 6 cm
(3) In triangle ABC right-angled at A,
Given AB = 6 cm and CA = 6 cm
BC
2
= AB
2
+ CA
2
{Using Pythagoras theorem}
? BC
2
= 6
2
+ 6
2
? BC
2
= 36 + 36
? BC = v72
Page 2
Circle
Practice Set 3.1
Q. 1. In the adjoining figure the radius of a circle with centre C is 6 cm, line AB is
a tangent at A. Answer the following questions.
(1) What is the measure of ?CAB? Why?
(2) What is the distance of point C from line AB? Why?
(3) d(A,B) = 6 cm, find d(B,C).
(4) What is the measure of ?ABC? Why?
Answer : (1) ere CA is the radius of the circle and A is the point of contact of the
tangent AB.
? ?CAB = 90° Using tangent-radius theorem which states that a tangent at any point of
a circle is perpendicular to the radius at the point of contact.
(2) CA is the radius of the circle which is perpendicular to the tangent AB.
So, the perpendicular distance of line AB from C = CA = 6 cm
(3) In triangle ABC right-angled at A,
Given AB = 6 cm and CA = 6 cm
BC
2
= AB
2
+ CA
2
{Using Pythagoras theorem}
? BC
2
= 6
2
+ 6
2
? BC
2
= 36 + 36
? BC = v72
? BC = 6v2 cm
(4) In triangle ABC right-angled at A,
AB = CA = 6 cm
??ABC = ?ACB {Angles opposite to equal sides are equal}
??ABC + ?ACB + ? BAC = 180° {Angle sum property of the triangle}
? 2 ?ABC = 90° { ? ? BAC = 90°}
? ?ABC = 45°
Q. 2. In the adjoining figure, O is the centre of the circle. From point R, seg RM
and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm
and radius of the circle = 5 cm, then
(1) What is the length of each tangent segment?
(2) What is the measure of ?MRO?
(3) What is the measure of ?MRN?
Answer : (1) Here OM is the radius of the circle and M and N are the points of contact
of MR and NR respectively.
? ?RMO = 90° Using tangent-radius theorem which states that a tangent at any point of
a circle is perpendicular to the radius at the point of contact.
In triangle ORM right-angled at M,
Given that OR = 10 cm and OM = 5 cm {Radius of the circle}
OR
2
= OM
2
+ RM
2
{Using Pythagoras theorem}
? MR
2
= 10
2
-5
2
Page 3
Circle
Practice Set 3.1
Q. 1. In the adjoining figure the radius of a circle with centre C is 6 cm, line AB is
a tangent at A. Answer the following questions.
(1) What is the measure of ?CAB? Why?
(2) What is the distance of point C from line AB? Why?
(3) d(A,B) = 6 cm, find d(B,C).
(4) What is the measure of ?ABC? Why?
Answer : (1) ere CA is the radius of the circle and A is the point of contact of the
tangent AB.
? ?CAB = 90° Using tangent-radius theorem which states that a tangent at any point of
a circle is perpendicular to the radius at the point of contact.
(2) CA is the radius of the circle which is perpendicular to the tangent AB.
So, the perpendicular distance of line AB from C = CA = 6 cm
(3) In triangle ABC right-angled at A,
Given AB = 6 cm and CA = 6 cm
BC
2
= AB
2
+ CA
2
{Using Pythagoras theorem}
? BC
2
= 6
2
+ 6
2
? BC
2
= 36 + 36
? BC = v72
? BC = 6v2 cm
(4) In triangle ABC right-angled at A,
AB = CA = 6 cm
??ABC = ?ACB {Angles opposite to equal sides are equal}
??ABC + ?ACB + ? BAC = 180° {Angle sum property of the triangle}
? 2 ?ABC = 90° { ? ? BAC = 90°}
? ?ABC = 45°
Q. 2. In the adjoining figure, O is the centre of the circle. From point R, seg RM
and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm
and radius of the circle = 5 cm, then
(1) What is the length of each tangent segment?
(2) What is the measure of ?MRO?
(3) What is the measure of ?MRN?
Answer : (1) Here OM is the radius of the circle and M and N are the points of contact
of MR and NR respectively.
? ?RMO = 90° Using tangent-radius theorem which states that a tangent at any point of
a circle is perpendicular to the radius at the point of contact.
In triangle ORM right-angled at M,
Given that OR = 10 cm and OM = 5 cm {Radius of the circle}
OR
2
= OM
2
+ RM
2
{Using Pythagoras theorem}
? MR
2
= 10
2
-5
2
? MR
2
= 100 - 25
? MR = v75
? MR = 5v3 cm
Also, RN = 5v3 cm { ? Tangents from the same external point are congruent to each
other.}
(2)
??MRO = 30°
(3) Similarly, ?NRO = 30°
??MRN = ? MRO + ?NRO = 30° + 30° = 60°
Q. 3. Seg RM and seg RN are tangent segments of a circle with centre O. Prove
that seg OR bisects ?MRN as well as ?MON.
Answer : In triangle MOR and triangle NOR,
MR = NR { ?Tangents from same external point are congruent to each other.}
OR = OR {Common}
OM = ON {Radius of the circle}
? ?MOR ? ?NOR {By SSS}
? ?ROM = ?RON
Page 4
Circle
Practice Set 3.1
Q. 1. In the adjoining figure the radius of a circle with centre C is 6 cm, line AB is
a tangent at A. Answer the following questions.
(1) What is the measure of ?CAB? Why?
(2) What is the distance of point C from line AB? Why?
(3) d(A,B) = 6 cm, find d(B,C).
(4) What is the measure of ?ABC? Why?
Answer : (1) ere CA is the radius of the circle and A is the point of contact of the
tangent AB.
? ?CAB = 90° Using tangent-radius theorem which states that a tangent at any point of
a circle is perpendicular to the radius at the point of contact.
(2) CA is the radius of the circle which is perpendicular to the tangent AB.
So, the perpendicular distance of line AB from C = CA = 6 cm
(3) In triangle ABC right-angled at A,
Given AB = 6 cm and CA = 6 cm
BC
2
= AB
2
+ CA
2
{Using Pythagoras theorem}
? BC
2
= 6
2
+ 6
2
? BC
2
= 36 + 36
? BC = v72
? BC = 6v2 cm
(4) In triangle ABC right-angled at A,
AB = CA = 6 cm
??ABC = ?ACB {Angles opposite to equal sides are equal}
??ABC + ?ACB + ? BAC = 180° {Angle sum property of the triangle}
? 2 ?ABC = 90° { ? ? BAC = 90°}
? ?ABC = 45°
Q. 2. In the adjoining figure, O is the centre of the circle. From point R, seg RM
and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm
and radius of the circle = 5 cm, then
(1) What is the length of each tangent segment?
(2) What is the measure of ?MRO?
(3) What is the measure of ?MRN?
Answer : (1) Here OM is the radius of the circle and M and N are the points of contact
of MR and NR respectively.
? ?RMO = 90° Using tangent-radius theorem which states that a tangent at any point of
a circle is perpendicular to the radius at the point of contact.
In triangle ORM right-angled at M,
Given that OR = 10 cm and OM = 5 cm {Radius of the circle}
OR
2
= OM
2
+ RM
2
{Using Pythagoras theorem}
? MR
2
= 10
2
-5
2
? MR
2
= 100 - 25
? MR = v75
? MR = 5v3 cm
Also, RN = 5v3 cm { ? Tangents from the same external point are congruent to each
other.}
(2)
??MRO = 30°
(3) Similarly, ?NRO = 30°
??MRN = ? MRO + ?NRO = 30° + 30° = 60°
Q. 3. Seg RM and seg RN are tangent segments of a circle with centre O. Prove
that seg OR bisects ?MRN as well as ?MON.
Answer : In triangle MOR and triangle NOR,
MR = NR { ?Tangents from same external point are congruent to each other.}
OR = OR {Common}
OM = ON {Radius of the circle}
? ?MOR ? ?NOR {By SSS}
? ?ROM = ?RON
And ?MRO = ?NRO {C.P.C.T.}
Hence proved that seg OR bisects ?MRNas well as ?MON.
Q. 4. What is the distance between two parallel tangents of a circle having
radius4.5 cm ? Justify your answer.
Answer : Let BC and DE be the parallel tangents to a circle centered at A with point of
contact O and H respectively. On joining OH, we find OH is the diameter of the
circle. ? BOA = 90° = ? DHA {Using tangent-radius theorem which states that a tangent
at any point of a circle is perpendicular to the radius at the point of contact.}
Distance between BC and DE = OH
? OH is perpendicular to BC and DE.
OH = 2 × 4.5 cm = 9 cm
Practice Set 3.2
Q. 1. Two circles having radii 3.5 cm and 4.8 cm touch each other internally. Find
the distance between their centres.
Answer : Given: Two circles are touching each other internally.
?The distance between the centres of the circles touching internally is equal to the
difference of their radii.
? Distance between their centres = 4.8 cm – 3.5 cm = 1.3 cm
Q. 2. Two circles of radii 5.5 cm and 4.2 cm touch each other externally. Find the
distance between their centres.
Page 5
Circle
Practice Set 3.1
Q. 1. In the adjoining figure the radius of a circle with centre C is 6 cm, line AB is
a tangent at A. Answer the following questions.
(1) What is the measure of ?CAB? Why?
(2) What is the distance of point C from line AB? Why?
(3) d(A,B) = 6 cm, find d(B,C).
(4) What is the measure of ?ABC? Why?
Answer : (1) ere CA is the radius of the circle and A is the point of contact of the
tangent AB.
? ?CAB = 90° Using tangent-radius theorem which states that a tangent at any point of
a circle is perpendicular to the radius at the point of contact.
(2) CA is the radius of the circle which is perpendicular to the tangent AB.
So, the perpendicular distance of line AB from C = CA = 6 cm
(3) In triangle ABC right-angled at A,
Given AB = 6 cm and CA = 6 cm
BC
2
= AB
2
+ CA
2
{Using Pythagoras theorem}
? BC
2
= 6
2
+ 6
2
? BC
2
= 36 + 36
? BC = v72
? BC = 6v2 cm
(4) In triangle ABC right-angled at A,
AB = CA = 6 cm
??ABC = ?ACB {Angles opposite to equal sides are equal}
??ABC + ?ACB + ? BAC = 180° {Angle sum property of the triangle}
? 2 ?ABC = 90° { ? ? BAC = 90°}
? ?ABC = 45°
Q. 2. In the adjoining figure, O is the centre of the circle. From point R, seg RM
and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm
and radius of the circle = 5 cm, then
(1) What is the length of each tangent segment?
(2) What is the measure of ?MRO?
(3) What is the measure of ?MRN?
Answer : (1) Here OM is the radius of the circle and M and N are the points of contact
of MR and NR respectively.
? ?RMO = 90° Using tangent-radius theorem which states that a tangent at any point of
a circle is perpendicular to the radius at the point of contact.
In triangle ORM right-angled at M,
Given that OR = 10 cm and OM = 5 cm {Radius of the circle}
OR
2
= OM
2
+ RM
2
{Using Pythagoras theorem}
? MR
2
= 10
2
-5
2
? MR
2
= 100 - 25
? MR = v75
? MR = 5v3 cm
Also, RN = 5v3 cm { ? Tangents from the same external point are congruent to each
other.}
(2)
??MRO = 30°
(3) Similarly, ?NRO = 30°
??MRN = ? MRO + ?NRO = 30° + 30° = 60°
Q. 3. Seg RM and seg RN are tangent segments of a circle with centre O. Prove
that seg OR bisects ?MRN as well as ?MON.
Answer : In triangle MOR and triangle NOR,
MR = NR { ?Tangents from same external point are congruent to each other.}
OR = OR {Common}
OM = ON {Radius of the circle}
? ?MOR ? ?NOR {By SSS}
? ?ROM = ?RON
And ?MRO = ?NRO {C.P.C.T.}
Hence proved that seg OR bisects ?MRNas well as ?MON.
Q. 4. What is the distance between two parallel tangents of a circle having
radius4.5 cm ? Justify your answer.
Answer : Let BC and DE be the parallel tangents to a circle centered at A with point of
contact O and H respectively. On joining OH, we find OH is the diameter of the
circle. ? BOA = 90° = ? DHA {Using tangent-radius theorem which states that a tangent
at any point of a circle is perpendicular to the radius at the point of contact.}
Distance between BC and DE = OH
? OH is perpendicular to BC and DE.
OH = 2 × 4.5 cm = 9 cm
Practice Set 3.2
Q. 1. Two circles having radii 3.5 cm and 4.8 cm touch each other internally. Find
the distance between their centres.
Answer : Given: Two circles are touching each other internally.
?The distance between the centres of the circles touching internally is equal to the
difference of their radii.
? Distance between their centres = 4.8 cm – 3.5 cm = 1.3 cm
Q. 2. Two circles of radii 5.5 cm and 4.2 cm touch each other externally. Find the
distance between their centres.
Answer : Given: Two circles are touching each other externally
We know that if the circles touch each other externally, distance between their centres
is equal to the sum of their radii.
? Distance between their centres = 5.5 cm + 4.2 cm = 9.7 cm
Q. 3. If radii of two circles are 4 cm and 2.8 cm. Draw figure of these circles
touching each other –
(i) externally
(ii) internally.
Answer :
Steps of construction:
1. Draw a circle with radius 4cm and centre A.
2. Draw another circle with radius 2.8 cm and centre B such that they touch each other
externally.
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FAQs on Textbook Solutions: Circle - Mathematics Class 10 (Maharashtra SSC Board)
| 1. What are the key properties of a circle that students should know for Class 10 exams? |  |
Ans. Students should be familiar with several key properties of circles, including:
1. The radius, which is the distance from the center of the circle to any point on its circumference.
2. The diameter, which is twice the radius and passes through the center of the circle, dividing it into two equal halves.
3. The circumference, which is the total distance around the circle, calculated using the formula C = 2πr, where r is the radius.
4. Chords, which are segments whose endpoints lie on the circle, and the longest chord is the diameter.
5. The relationship between angles, such as the angle subtended by a chord at the center being twice the angle subtended at any point on the remaining part of the circumference.
| 2. How do you calculate the area and circumference of a circle? | |
Ans. The area and circumference of a circle can be calculated using the following formulas:
- Circumference (C) = 2πr, where r is the radius of the circle. This gives the total distance around the circle.
- Area (A) = πr², where r is again the radius. This formula provides the total space enclosed within the circle.
| 3. What is the significance of theorems related to circles in Class 10 mathematics? | |
Ans. Theorems related to circles are crucial as they help in understanding the geometric properties and relationships within circles and between circles and other shapes. For example, theorems about angles subtended by chords, the relationship between tangents and radii, and the properties of cyclic quadrilaterals are essential for solving various geometric problems. Mastering these theorems not only aids in exam preparation but also enhances overall mathematical reasoning.
| 4. Can you explain the concept of tangents to a circle? | |
Ans. A tangent to a circle is a straight line that touches the circle at exactly one point, known as the point of tangency. Some important properties of tangents include that the radius drawn to the point of tangency is perpendicular to the tangent line. Additionally, if two tangents are drawn from an external point to a circle, they will be equal in length. Understanding tangents is important for solving problems related to circles in geometry.
| 5. What are some common mistakes students make when studying circles? | |
Ans. Common mistakes include:
1. Miscalculating the radius or diameter, which can lead to incorrect area or circumference calculations.
2. Confusing the properties of chords and tangents, especially regarding angles.
3. Neglecting to apply theorems correctly, which can result in errors in geometric proofs.
4. Failing to label diagrams accurately, making it hard to visualize and solve problems effectively.
5. Not practicing enough with problems related to circles, which can lead to a lack of familiarity with the concepts.
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