Class 9 NCERT Solutions Maths Chapter 9 - Circles
Ex. 9.1
Ex. 9.2
Ex. 9.3
Circles Exercise Ex. 9.1
Solution 1
A circle is a collection of points which are equidistant from a fix point. This fix point is called as the centre of circle and this equal distance is called as radius of circle. And thus shape of a circle depends on the radius of the circle.
So, if we try to superimpose two circles of equal radius, one each other both circles will cover each other.
So, two circles are congruent if they have equal radius.
So, if we try to superimpose two circles of equal radius, one each other both circles will cover each other.
So, two circles are congruent if they have equal radius.
Now consider two congruent circles having centre O and O' and two chords AB and CD of equal lengths
Now in
AOB and
CO'D
AB = CD (chords of same length)
OA = O'C (radii of congruent circles)
OB = O'D (radii of congruent circles)
AOB
CO'D (SSS congruence rule)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1134_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1134_ec7fd5b2b4109644a398eba4d25ebf00.png)
AB = CD (chords of same length)
OA = O'C (radii of congruent circles)
OB = O'D (radii of congruent circles)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1134_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1134_3048037af7aaa664a9d374c13f47ecfe.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1134_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1134_13fb2c7abb7fb90f3d38dc838d646ea3.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1134_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1134_4934c411dbc1a55c0b7875b3e0fc862a.png)
Hence equal chords of congruent circles subtend equal angles at their centres.
Solution 2
Let us consider two congruent circles (circles of same radius) with centres as O and O'.
In
AOB and
CO'D
AOB =
CO'D (given)
OA = O'C (radii of congruent circles)
OB = O'D (radii of congruent circles)
![](https://images.topperlearning.com/topper/bookquestions/1136_im19.JPG)
AOB
CO'D (SSS congruence rule)
AB = CD (by CPCT)
Hence, if chords of congruent circles subtend equal angles at their centres then chords are equal.
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1136_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1136_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1136_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1136_4934c411dbc1a55c0b7875b3e0fc862a.png)
OA = O'C (radii of congruent circles)
OB = O'D (radii of congruent circles)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1136_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1136_6cf26576c060f5f190df510e5535ff3c.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1136_13fb2c7abb7fb90f3d38dc838d646ea3.png)
Hence, if chords of congruent circles subtend equal angles at their centres then chords are equal.
Circles Exercise Ex. 9.2
Solution 1
Let radius of circle centered at O and O' be 5 cm and 3 cm respectively.
OA = OB = 5 cm
O'A = O'B = 3 cm
OO' will be the perpendicular bisector of chord AB.
AC = CB
OA = OB = 5 cm
O'A = O'B = 3 cm
OO' will be the perpendicular bisector of chord AB.
Given that OO' = 4 cm
Let OC be x. so, O'C will be 4 - x
In
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1143_ec7fd5b2b4109644a398eba4d25ebf00.png)
OA2 = AC2 + OC2
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1143_13fb2c7abb7fb90f3d38dc838d646ea3.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1143_13fb2c7abb7fb90f3d38dc838d646ea3.png)
In
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1143_ec7fd5b2b4109644a398eba4d25ebf00.png)
O'A2 = AC2 + O'C2
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1143_13fb2c7abb7fb90f3d38dc838d646ea3.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1143_13fb2c7abb7fb90f3d38dc838d646ea3.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1143_13fb2c7abb7fb90f3d38dc838d646ea3.png)
From equations (1) and (2), we have
25 - x2 = - x2 - 7 + 8x
8x = 32
x = 4
So, the common chord will pass through the centre of smaller circle i.e. O'. and hence it will be diameter of smaller circle.
Now, AC2 = 25 - x2 = 25 - 42 = 25 - 16 = 9
AC = 3 m
The length of the common chord AB = 2 AC = (2
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1143_b9ccb383b0c73c26cfb3ff194a86e235.png)
Solution 2
Let PQ and RS are two equal chords of a given circle and there are intersecting each other at point T.
Draw perpendiculars OV and OU on these chords.
In
OVT and
OUT
OV = OU (Equal chords of a circle are equidistant from the centre)
OVT =
OUT (Each 90o)
OT = OT (common)
In
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1144_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1144_ec7fd5b2b4109644a398eba4d25ebf00.png)
OV = OU (Equal chords of a circle are equidistant from the centre)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1144_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1144_4934c411dbc1a55c0b7875b3e0fc862a.png)
OT = OT (common)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1144_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1144_3048037af7aaa664a9d374c13f47ecfe.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1144_ec7fd5b2b4109644a398eba4d25ebf00.png)
It is given that
PQ = RS ... ... ... ... (2)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1144_13fb2c7abb7fb90f3d38dc838d646ea3.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1144_cb3cfe781f270733f4acbe2739f2e845.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1144_13fb2c7abb7fb90f3d38dc838d646ea3.png)
On adding equations (1) and (3), we have
PV + VT = RU + UT
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1144_13fb2c7abb7fb90f3d38dc838d646ea3.png)
On subtracting equation (4) from equation (2), we have
PQ - PT = RS - RT
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1144_13fb2c7abb7fb90f3d38dc838d646ea3.png)
Equations (4) and (5) shows that the corresponding segments of
chords PQ and RS are congruent to each other.
Solution 3
Let PQ and RS are two equal chords of a given circle and there are intersecting each other at point T.
Draw perpendiculars OV and OU on these chords.
In
OVT and
OUT
OV = OU (Equal chords of a circle are equidistant from the centre)
OVT =
OUT (Each 90o)
OT = OT (common)
![](https://images.topperlearning.com/topper/bookquestions/1146_im19.JPG)
OVT
OUT (RHS congruence rule)
![](https://images.topperlearning.com/topper/bookquestions/1146_im19.JPG)
OTV =
OTU (by CPCT)
Draw perpendiculars OV and OU on these chords.
In
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1146_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1146_ec7fd5b2b4109644a398eba4d25ebf00.png)
OV = OU (Equal chords of a circle are equidistant from the centre)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1146_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1146_4934c411dbc1a55c0b7875b3e0fc862a.png)
OT = OT (common)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1146_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1146_1761c87e2ffbc82d2d19eafa2b97f1ee.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1146_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1146_4934c411dbc1a55c0b7875b3e0fc862a.png)
Hence, the line joining the point of intersection to the centre makes equal angles with the chords.
Solution 4
Let us draw a perpendicular OM on line AD.
![](https://images.topperlearning.com/topper/bookquestions/1147_ss38.JPG)
Here, BC is chord of smaller circle and AD is chord of bigger circle.
We know that the perpendicular drawn from centre of circle bisects the chord.
And AM = MD ... (2)
Subtracting equations (2) from (1), we have
AM - BM = MD - MC
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1147_13fb2c7abb7fb90f3d38dc838d646ea3.png)
Solution 5
Draw perpendiculars OA and OB on RS and SM respectively.
Let R, S and M be the position of Reshma, Salma and Mandip respectively.
Let R, S and M be the position of Reshma, Salma and Mandip respectively.
AR = AS =
= 3cm
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1148_e7138cd17cc5cbab052d07b344a6561a.png)
OR = OS = OM = 5 m (radii of circle)
In OAR
OA2 + AR2 = OR2
OA2 + (3 m)2 = (5 m)2
OA2 = (25 - 9) m2 = 16 m2
OA = 4 m
We know that in an isosceles triangle altitude divides the base, so in
RSM
RCS will be of 90o and RC = CM
Area of
ORS = ![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1148_fcdbbc6bbc2ca084b11561f01b4ff909.png)
OA
RS
In OAR
OA2 + AR2 = OR2
OA2 + (3 m)2 = (5 m)2
OA2 = (25 - 9) m2 = 16 m2
OA = 4 m
We know that in an isosceles triangle altitude divides the base, so in
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1148_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1148_4934c411dbc1a55c0b7875b3e0fc862a.png)
Area of
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1148_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1148_fcdbbc6bbc2ca084b11561f01b4ff909.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1148_b9ccb383b0c73c26cfb3ff194a86e235.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1148_b9ccb383b0c73c26cfb3ff194a86e235.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1148_58278dcee5e9af75a501682ad57677c6.png)
RC = 4.8
RM = 2RC = 2(4.8)= 9.6
So, distance between Reshma and Mandip is 9.6 m.
Solution 6
Given that AS = SD = DA
So, ASD is a equilateral triangle
OA (radius) = 20 m.
Medians of equilateral triangle pass through the circum centre (O) of the equilateral triangle ABC.
We also know that median intersect each other at the 2: 1. As AB is the median of equilateral triangle ABC, we can write
So, ASD is a equilateral triangle
OA (radius) = 20 m.
Medians of equilateral triangle pass through the circum centre (O) of the equilateral triangle ABC.
We also know that median intersect each other at the 2: 1. As AB is the median of equilateral triangle ABC, we can write
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1149_4fc757dc1843694979d0707183e946ce.png)
In
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1149_ec7fd5b2b4109644a398eba4d25ebf00.png)
AD2 = AB2 + BD2
AD2 = (30)2 +
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1149_05a5194335b1c58ef14cc1f0b10e0df9.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1149_4564a707c38ef19c8e27d8788208caea.png)
So, length of string of each phone will be
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1149_c5c7af935a78ade28040a27273ff50e4.png)
Circles Exercise Ex. 9.3
Solution 1
We may observe that
AOC =
AOB +
BOC
= 60o + 30o
= 90o
We know that angle subtended by an arc at centre is double the angle subtended by it any point on the remaining part of the circle.
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1151_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1151_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1151_4934c411dbc1a55c0b7875b3e0fc862a.png)
= 60o + 30o
= 90o
We know that angle subtended by an arc at centre is double the angle subtended by it any point on the remaining part of the circle.
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1151_21b105174c84aaff394604cd4649cc6b.png)
Solution 2
In
OAB
AB = OA = OB = radius
![](https://images.topperlearning.com/topper/bookquestions/1153_im19.JPG)
OAB is an equilateral triangle.
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1153_ec7fd5b2b4109644a398eba4d25ebf00.png)
AB = OA = OB = radius
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1153_ec7fd5b2b4109644a398eba4d25ebf00.png)
So, each interior angle of this triangle will be of 60o
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1153_4934c411dbc1a55c0b7875b3e0fc862a.png)
Now, ![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1153_bd4e9e438b3f4f6a0cdd95b6666f734c.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1153_bd4e9e438b3f4f6a0cdd95b6666f734c.png)
In cyclic quadrilateral ACBD
ACB +
ADB = 180o (Opposite angle in cyclic quadrilateral)
ADB = 180o - 30o = 150o
So, angle subtended by this chord at a point on major arc and minor arc are 30o and 150o respectively.
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1153_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1153_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1153_2457d20c0787e1fb421665bd02d123a1.png)
So, angle subtended by this chord at a point on major arc and minor arc are 30o and 150o respectively.
Solution 3
Consider PR as a chord of circle.
Take any point S on major arc of circle.
Now PQRS is a cyclic quadrilateral.
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1155_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1155_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1155_86e9f4603b1f61504655f58bc01b32c4.png)
We know that angle subtended by an arc at centre is double the angle subtended by it any point on the remaining part of the circle.
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1155_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1155_4934c411dbc1a55c0b7875b3e0fc862a.png)
In
POR
OP = OR (radii of same circle)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1155_ec7fd5b2b4109644a398eba4d25ebf00.png)
OP = OR (radii of same circle)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1155_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1155_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1155_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1155_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1155_4934c411dbc1a55c0b7875b3e0fc862a.png)
2
OPR + 160o= 180o
2
OPR = 180o - 160o = 20o
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1155_4934c411dbc1a55c0b7875b3e0fc862a.png)
2
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1155_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1155_4934c411dbc1a55c0b7875b3e0fc862a.png)
Solution 4
In
ABC
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1156_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1156_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1156_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1156_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1156_2457d20c0787e1fb421665bd02d123a1.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1156_2457d20c0787e1fb421665bd02d123a1.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1156_2457d20c0787e1fb421665bd02d123a1.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1156_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1156_4934c411dbc1a55c0b7875b3e0fc862a.png)
Solution 5
In
CDE
CDE +
DCE =
CEB (Exterior angle)
CDE + 20o = 130o
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1158_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1158_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1158_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1158_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1158_2457d20c0787e1fb421665bd02d123a1.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1158_2457d20c0787e1fb421665bd02d123a1.png)
But
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1158_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1158_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1158_2457d20c0787e1fb421665bd02d123a1.png)
Solution 6
For chord CD
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
In
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_ec7fd5b2b4109644a398eba4d25ebf00.png)
AB = BC (given)
![](https://images.topperlearning.com/topper/bookquestions/1159_im19.JPG)
BCA =
CAB (Angles opposite to equal sides of a triangle)
BCA = 30o
We have
BCD = 80o
BCA +
ACD = 80o
30o +
ACD = 80o
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
We have
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
30o +
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1159_4934c411dbc1a55c0b7875b3e0fc862a.png)
Solution 7
Let ABCD a cyclic quadrilateral having diagonals as BD and AC intersecting each other at point O.
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1160_3722fd1f766e804fad6fc84694ed27ad.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1160_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1160_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1160_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1160_5ef77821ce67cada25c8018afa0461f7.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1160_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1160_4934c411dbc1a55c0b7875b3e0fc862a.png)
90o +
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1160_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1160_4934c411dbc1a55c0b7875b3e0fc862a.png)
Here, each interior angle of cyclic quadrilateral is of 90o. Hence it is a rectangle.
Solution 8
Consider a trapezium ABCD with AB | |CD and BC = AD Draw AM
CD and BN
CD
In
AMD and
BNC
AD = BC (Given)
AMD =
BNC (By construction each is 90o)
AM = BM (Perpendicular distance between two parallel lines is same)
![](https://images.topperlearning.com/topper/bookquestions/1164_im19.JPG)
AMD
BNC (RHS congruence rule)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_cd0ab85520b5ae722383bdcc2bb17bd8.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_cd0ab85520b5ae722383bdcc2bb17bd8.png)
In
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_ec7fd5b2b4109644a398eba4d25ebf00.png)
AD = BC (Given)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_4934c411dbc1a55c0b7875b3e0fc862a.png)
AM = BM (Perpendicular distance between two parallel lines is same)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_6cf26576c060f5f190df510e5535ff3c.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1164_4934c411dbc1a55c0b7875b3e0fc862a.png)
This equation shows that the opposite angles are supplementary.
So, ABCD is a cyclic quadrilateral.
Solution 9
![](https://images.topperlearning.com/topper/bookquestions/1165_Ex_10.5_9A.png)
Join chords AP and DQ
For chord AP
PBA =
ACP (Angles in same segment) ... (1)
For chord DQ
DBQ =
QCD (Angles in same segment) ... (2)
ABD and PBQ are line segments intersecting at B.
![](https://images.topperlearning.com/topper/bookquestions/1165_im19.JPG)
PBA =
DBQ (Vertically opposite angles) ... (3)
From equations (1), (2) and (3), we have
ACP =
QCD
For chord AP
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1165_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1165_4934c411dbc1a55c0b7875b3e0fc862a.png)
For chord DQ
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1165_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1165_4934c411dbc1a55c0b7875b3e0fc862a.png)
ABD and PBQ are line segments intersecting at B.
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1165_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1165_4934c411dbc1a55c0b7875b3e0fc862a.png)
From equations (1), (2) and (3), we have
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1165_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1165_4934c411dbc1a55c0b7875b3e0fc862a.png)
Solution 10
Consider a
ABC
Two circles are drawn while taking AB and AC as diameter.
Let they intersect each other at D and let D does not lie on BC.
Join AD
ADB = 90o (Angle subtend by semicircle)
ADC = 90o (Angle subtend by semicircle)
BDC =
ADB +
ADC = 90o + 90o = 180o
Hence BDC is straight line and our assumption was wrong.
Thus, Point D lies on third side BC of
ABC
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1166_ec7fd5b2b4109644a398eba4d25ebf00.png)
Two circles are drawn while taking AB and AC as diameter.
Let they intersect each other at D and let D does not lie on BC.
Join AD
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1166_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1166_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1166_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1166_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1166_4934c411dbc1a55c0b7875b3e0fc862a.png)
Hence BDC is straight line and our assumption was wrong.
Thus, Point D lies on third side BC of
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1166_ec7fd5b2b4109644a398eba4d25ebf00.png)
Solution 11
In
ABC
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_13fb2c7abb7fb90f3d38dc838d646ea3.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_13fb2c7abb7fb90f3d38dc838d646ea3.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
In
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_ec7fd5b2b4109644a398eba4d25ebf00.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_13fb2c7abb7fb90f3d38dc838d646ea3.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_13fb2c7abb7fb90f3d38dc838d646ea3.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
Adding equations (1) and (2), we have
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_13fb2c7abb7fb90f3d38dc838d646ea3.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
But it is given that
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
From equations (3) and (4), we can see that quadrilateral ABCD is having sum of measures of opposite angles as 180o.
So, it is a cyclic quadrilateral.
Consider chord CD.
Now,
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1168_4934c411dbc1a55c0b7875b3e0fc862a.png)
Solution 12
Let ABCD be a cyclic parallelogram.
A +
C = 180o (Opposite angle of cyclic quadrilateral) ... (1)
We know that opposite angles of a parallelogram are equal
![](https://images.topperlearning.com/topper/bookquestions/1169_im19.JPG)
A =
C and
B =
D
From equation (1)
A +
C = 180o
A +
A = 180o
2
A = 180o
A = 90o
Parallelogram ABCD is having its one of interior angles as 90o, so, it is a rectangle.
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1169_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1169_4934c411dbc1a55c0b7875b3e0fc862a.png)
We know that opposite angles of a parallelogram are equal
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1169_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1169_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1169_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1169_4934c411dbc1a55c0b7875b3e0fc862a.png)
From equation (1)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1169_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1169_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1169_86e9f4603b1f61504655f58bc01b32c4.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1169_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1169_13fb2c7abb7fb90f3d38dc838d646ea3.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1169_4934c411dbc1a55c0b7875b3e0fc862a.png)
![Double click to edit](https://images.topperlearning.com/topper/bookquestions/1169_86e9f4603b1f61504655f58bc01b32c4.png)
Parallelogram ABCD is having its one of interior angles as 90o, so, it is a rectangle.