Access Answers for SAT MATHS Chapter 10 Circles
Exercise: 10.1
1. Fill in the blanks:
(i) The centre of a circle lies in ____________ of the circle. (exterior/ interior)
(ii) A point, whose distance from the centre of a circle is greater than its radius lies in __________ of
the circle. (exterior/ interior)
(iii) The longest chord of a circle is a _____________ of the circle.
(iv) An arc is a ___________ when its ends are the ends of a diameter.
(v) Segment of a circle is the region between an arc and _____________ of the circle.
(vi) A circle divides the plane, on which it lies, in _____________ parts.
Solution:
(i) The centre of a circle lies in interior of the circle.
(ii) A point, whose distance from the centre of a circle is greater than its radius lies in exterior of the
circle.
(iii) The longest chord of a circle is a diameter of the circle.
(iv) An arc is a semicircle when its ends are the ends of a diameter.
(v) Segment of a circle is the region between an arc and chord of the circle.
(vi) A circle divides the plane, on which it lies, in 3 (three) parts.
2. Write True or False: Give reasons for your Solutions.
(i) Line segment joining the centre to any point on the circle is a radius of the circle.
(ii) A circle has only finite number of equal chords.
(iii) If a circle is divided into three equal arcs, each is a major arc.
(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.
(v) Sector is the region between the chord and its corresponding arc.
(vi) A circle is a plane figure.
Solution:
(i) True. Any line segment drawn from the centre of the circle to any point on it is the radius of the
circle and will be of equal length.
(ii) False. There can be infinite numbers of equal chords of a circle.
(iii) False. For unequal arcs, there can be major and minor arcs. So, equal arcs on a circle cannot be
said as a major arc or a minor arc.
(iv) True. Any chord whose length is twice as long as the radius of the circle always passes through
the centre of the circle and thus, it is known as the diameter of the circle.
(v) False. A sector is a region of a circle between the arc and the two radii of the circle.
,(vi) True. A circle is a 2d figure and it can be drawn on a plane.
Exercise: 10.2
1. Recall that two circles are congruent if they have the same radii. Prove that equal chords of
congruent circles subtend equal angles at their centres.
Solution:
To recall, a circle is a collection of points whose every point is equidistant from its centre. So, two
circles can be congruent only when the distance of every point of both the circles are equal from the
centre.
For the second part of the question, it is given that AB = CD i.e. two equal chords.
Now, it is to be proven that angle AOB is equal to angle COD.
Proof:
Consider the triangles ΔAOB and ΔCOD,
OA = OC and OB = OD (Since they are the radii of the circle)
AB = CD (As given in the question)
So, by SSS congruency, ΔAOB ≅ ΔCOD
∴ By CPCT we have,
∠AOB = ∠COD. (Hence proved).
2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords
are equal.
Solution:
Consider the following diagram-
, Here, it is given that ∠AOB = ∠COD i.e. they are equal angles.
Now, we will have to prove that the line segments AB and CD are equal i.e. AB = CD.
Proof:
In triangles AOB and COD,
∠AOB = ∠COD (as given in the question)
OA = OC and OB = OD (these are the radii of the circle)
So, by SAS congruency, ΔAOB ≅ ΔCOD.
∴ By the rule of CPCT, we have
AB = CD. (Hence proved).
Exercise: 10.3
1. Draw different pairs of circles. How many points does each pair have in common? What is the
maximum number of common points?
Solution:
In these two circles, no point is common.
Exercise: 10.1
1. Fill in the blanks:
(i) The centre of a circle lies in ____________ of the circle. (exterior/ interior)
(ii) A point, whose distance from the centre of a circle is greater than its radius lies in __________ of
the circle. (exterior/ interior)
(iii) The longest chord of a circle is a _____________ of the circle.
(iv) An arc is a ___________ when its ends are the ends of a diameter.
(v) Segment of a circle is the region between an arc and _____________ of the circle.
(vi) A circle divides the plane, on which it lies, in _____________ parts.
Solution:
(i) The centre of a circle lies in interior of the circle.
(ii) A point, whose distance from the centre of a circle is greater than its radius lies in exterior of the
circle.
(iii) The longest chord of a circle is a diameter of the circle.
(iv) An arc is a semicircle when its ends are the ends of a diameter.
(v) Segment of a circle is the region between an arc and chord of the circle.
(vi) A circle divides the plane, on which it lies, in 3 (three) parts.
2. Write True or False: Give reasons for your Solutions.
(i) Line segment joining the centre to any point on the circle is a radius of the circle.
(ii) A circle has only finite number of equal chords.
(iii) If a circle is divided into three equal arcs, each is a major arc.
(iv) A chord of a circle, which is twice as long as its radius, is a diameter of the circle.
(v) Sector is the region between the chord and its corresponding arc.
(vi) A circle is a plane figure.
Solution:
(i) True. Any line segment drawn from the centre of the circle to any point on it is the radius of the
circle and will be of equal length.
(ii) False. There can be infinite numbers of equal chords of a circle.
(iii) False. For unequal arcs, there can be major and minor arcs. So, equal arcs on a circle cannot be
said as a major arc or a minor arc.
(iv) True. Any chord whose length is twice as long as the radius of the circle always passes through
the centre of the circle and thus, it is known as the diameter of the circle.
(v) False. A sector is a region of a circle between the arc and the two radii of the circle.
,(vi) True. A circle is a 2d figure and it can be drawn on a plane.
Exercise: 10.2
1. Recall that two circles are congruent if they have the same radii. Prove that equal chords of
congruent circles subtend equal angles at their centres.
Solution:
To recall, a circle is a collection of points whose every point is equidistant from its centre. So, two
circles can be congruent only when the distance of every point of both the circles are equal from the
centre.
For the second part of the question, it is given that AB = CD i.e. two equal chords.
Now, it is to be proven that angle AOB is equal to angle COD.
Proof:
Consider the triangles ΔAOB and ΔCOD,
OA = OC and OB = OD (Since they are the radii of the circle)
AB = CD (As given in the question)
So, by SSS congruency, ΔAOB ≅ ΔCOD
∴ By CPCT we have,
∠AOB = ∠COD. (Hence proved).
2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords
are equal.
Solution:
Consider the following diagram-
, Here, it is given that ∠AOB = ∠COD i.e. they are equal angles.
Now, we will have to prove that the line segments AB and CD are equal i.e. AB = CD.
Proof:
In triangles AOB and COD,
∠AOB = ∠COD (as given in the question)
OA = OC and OB = OD (these are the radii of the circle)
So, by SAS congruency, ΔAOB ≅ ΔCOD.
∴ By the rule of CPCT, we have
AB = CD. (Hence proved).
Exercise: 10.3
1. Draw different pairs of circles. How many points does each pair have in common? What is the
maximum number of common points?
Solution:
In these two circles, no point is common.
