Page 12.50

### EXERCISE 12.2

**Type 1**

#### QUESTION 1

**(i) AB is the chord of a circle. If P and Q are two points in the same segment and ∠APB=48° , then which of the following is the measure of ∠AQB ?**

(a) 42°

(b) 48°

(c) 132°

(d) 30°

**(ii) AB is the diameter of a circle and C is a point on circle not one AB , then which of the following is the measure of ∠ACB ?**

(a) 180°

(b) 90°

(c) 0°

(d) 60°

**(iii) An angle of a cyclic quadrilateral is 70° , what will be value of the opposite angle ?**

(a) 30°

(b) 90°

(c) 20°

(d) 110°

**(iv) If opposite angles of a quadrilateral are 72° and 108° , then this opposite angle ?**

(a) parallelogram

(b) rhombus

(c) cyclic quadrilateral

(d) none of these

**(v) Angle subtended by an arc of a circle at its centre is 80°. What will be the the angle subtended by the same arc at any point on the remaining part of the circle ?**

(a) 100°

(b) 10°

(c) 40°

(d) 160°

**(vi) In a circle quadrilateral ABCD , ∠A=40° , ∠B=50° , then the value of opposite angles are :**

(a) 140° , 130°

(b) 150° , 140°

(c) 50° , 40°

(d) 40° , 150°

**(vii) A hexagon is inscribed in a circle , what angle will be subtended at the centre by each side ?**

(a) 90°

(b) 120°

(c) 60°

(d) 45°

**(viii) An angle of a cyclic quadrilateral is 75° , what will be the value of the opposite angle ?**

(a) 15°

(b) 25°

(c) 90°

(d) 105°

**(ix) In equal circles , the ratio of length of corresponding chords of equal arcs is :**

(a) 1:1

(b) 1:2

(c) 1:3

(d) none of these

Sol :

#### QUESTION 2

(i) Angle in a semicircle is __

<fig to be added>

(ii) In the given figure (i) , if ∠ABC=40° , then ∠ADC=__

<fig to be added>

(iii) In the given figure (ii) , if O is the centre of the circle, then ∠CDA=__

<fig to be added>

Sol :

#### QUESTION 3

**(i) In the given figure (iii), ∠BAC=70° , then write the value of ∠BDC .**

**(ii) If in a cyclic quadrilateral ABCD, ∠ADC=120° , then find the value of ∠ABC**

**(iii) If the chord of a segment of a circle subtends an angle 120° at the centre. What will be the measure of the angle subtended on the arc of the same segment ?**

**(iv) How many right angles are each angle of a cyclic parallelogram ? **

**(v) In a quadrilateral ABCD , ∠A=110° and ∠C=70° . How is this quadrilateral ?**

**(vi) How many right angles is the sum of opposite angles of a cyclic quadrilateral ?**

**(vii) The side DA of a cyclic quadrilateral is produced to point E . If ∠BAE=56° . What will be the measure of ∠BCD in degrees ?**

**(viii) What is the relation between an exterior angle formed when one side of a cyclic quadrilateral is produced and opposite interior angle ?**

**(ix) What is the shape of a cyclic rhombus ?**

**(x) What kind of angle is formed in the major segment ?**

**(xi) At what point, the circle drawn with one of the equal sides of an isosceles triangle as diameter will cut the base of the triangle ?**

**(xii) Hypotenuse of a right angled triangle is 10 cm . What will be the length of the line segment joining the mid-point of hypotenuse to right angle vertex ?**

Sol :

#### QUESTION 4

**(i) In the figure, ∠ABC=40° , if ∠DAC=65° , then ∠DCA=__**

<fig to be added>

**(ii) In the figure , if O be the centre of the circle , then ∠PQR=__**

<fig to be added>

**(iii) In the given figure , ∠AOC=__**

<fig to be added>

Sol :

#### QUESTION 5

**In the given figure, if ∠A=50° and ∠DCP=20° , then find ∠BDC and ∠DPC**

<fig to be added>

Sol :

#### QUESTION 6

**In the following figure, O is the centre of the circle and ∠BAC=30° , write the measure of the following angles :**

(i) ∠BOC

(ii) ∠CDE

<fig to be added>

Sol :

#### QUESTION 7

**(i) In figure (i) , A , B and C sre three points on the circle with centre O . AB and AC are two chords which subtend 90° , and 110° at the centre . Determine ∠BAC**

<fig to be added>

**(ii) In figure (ii) , O is the centre of the circle ABC. Find ∠BAC .**

Sol :

#### QUESTION 8

**AB is the diameter of a circle and P is any point on the circumference of the circle. If ∠PBA=40° , then find ∠PAB in degrees .**

Sol :

#### QUESTION 9

**A chord of a circle is equal to the radius of the circle . Find the measure of the angles formed by the chord on the circumference of the major segment .**

Sol :

**TYPE 2**

#### QUESTION 10

**If two opposite angles of a cyclic quadrilateral are in the ratio 5:7 , find the value of two angles in degree.**

Sol :

#### QUESTION 11

**PQRS is a cyclic quadrilateral in a circle with centre O which is on PR and ∠QSR=48° , then find the value of ∠PRQ**

Sol :

#### QUESTION 12

In figure (i) , O is the centre of the circle and ∠BOC=140° then write the measures of the following :

<fig to be added>

(i) ∠BDC=x

(ii) ∠BAC=y

(iii) In figure (ii) , ΔABC is an equilateral. Find ∠BDC and ∠BEC

<fig to be added>

Sol :

#### QUESTION 13

**In figure (i) , (ii) and (iii) given below . Find the angles x and y**

<fig to be added>

Sol :

#### QUESTION 14

**ABCD is a cyclic rhombus in which AD||BC and ∠B=70° , then find the Other three angles.**

Sol :

#### QUESTION 15

**The circumference of a circle is divided in the ratio 2:3:4 . Find the angles of the triangle obtained on joining the dividing points**

Sol :

#### QUESTION 16

**If in a triangle ABC , ∠A=71° and perpendiculars from vertices B and C on the opposite sides intersect at point P , find ∠BPC**

Sol :

#### QUESTION 17

**(i) In figure (i) lines AB and CD pass through the centre of the circle . If ∠AOC=80° , ∠CDB=40° , then find**

(a) ∠DBC

(b) ∠ABC

<fig to be added>

**(ii) In figure (ii) , find angle x**

<fig to be added>

**(iii) In figure (iii) , O is the centre of the circle. Find ∠BOC**

<fig to be added>

**(iv) In figure (iv) , O is the centre of the circle and ∠BDC=42° , find ∠ACB**

<fig to be added>

**(v) In figure (v) , ABCD is a cyclic quadrilateral whose side AB is the diameter of circle through the vertices A , B , C and D . If ∠ADC=140° then find ∠BAC**

<fig to be added>

Sol :

**TYPE 3**

#### QUESTION 18

**If all the sides of a cyclic quadrilateral are equal, then prove that the quadrilateral must be a square.**

Sol :

#### QUESTION 19

**Two diameters of a circle intersect each other at right angles. Prove that the quadrilateral formed by joining their end points is a square.**

Sol :

#### QUESTION 20

**Prove that each square and rectangle is a cyclic quadrilateral. **

Sol :

#### QUESTION 21

BC is the chord of the circle whose centre is O . A is any point lying on the arc , as shown in figures (i) and (ii) . Prove that

<fig to be added>

(i) ∠BAC+∠OBC=90° , when point A lies on the , major arc

(ii) ∠BAC-∠OBC=90° , when point A lies on the minor arc

Sol :

#### QUESTION 22

**In the figure, AB and CD are diameters of the circle C(O,r) . Prove that AC||BD and AD||BC . If ∠OBD=50° , find ∠AOC**

<fig>

<hint>

Sol :

#### QUESTION 23

In the figure , PQ and RQ are two chords of a circle with arc equidistant from centre O . Prove that the diameter QS bisects ∠PQR and ∠PSR

<fig>

Sol :

#### QUESTION 24

In the figure , ABCD is a cyclic quadrilateral . Circle passing through A and B , intersect AD and BC at points E and F respectively. Prove that EF||DC .

<fig>

<hint>

Sol :

#### QUESTION 25

**Two circles intersect at points A and D and AC and AD are the diameters of these circles . Prove that B , C and D are collinear**

Sol :

#### QUESTION 26

**Prove that , a circle drawn taking one of the equal sides of an isosceles triangle as diameter bisects the base of the triangle.**

<fig>

<hint>

Sol :

#### QUESTION 27

**In the figure , ABC is a triangle in which AB = AC and circle Passing through B and C intersects sides AB and AC at point D and E respectively . Prove that DE||BC**

<fig>

<Hint>

Sol :

#### QUESTION 28

**AB and AC are two equal sides of the triangle ABC on which two points D and E respectively are such that AD=AE . Prove that point B , C , E and D are concyclic**

<fig>

<Hint>

Sol :

#### QUESTION 29

**Prove that perpendicular bisectors of sides of a cyclic quadrilateral are concurrent **

<fig>

<Hint>

Sol :

#### QUESTION 30

**If two sides of a cyclic quadrilateral are parallel, then prove that **

**(i) remaining two sides are equal **

**(ii) both diagonals are equal**

<fig>

<Hint>

Sol :

#### QUESTION 31

**If two non-parallel sides of a trapezium are equal , prove that it is a cyclic quadrilateral **

<fig>

<Hint>

Sol :

#### QUESTION 32

**In cyclic quadrilateral ABCD , opposite sides AB and DC , on being produced intersect at E . Prove that ΔEAD~ΔECB**

<fig>

<Hint>

Sol :

#### QUESTION 33

**From point of intersection A of the two congruent circles , line segments MAN and RAS are drawn . Prove that chords MR and NS are equal**

<fig>

<Hint>

Sol :

#### QUESTION 34

**Two circles are drawn taking any two sides of a triangle as diameter . Prove that the circles intersect on the third side (or on produced third side)**

<fig>

<Hint>

Sol :

#### QUESTION 35

**Prove that two opposite arcs formed between two parallel chords of a circle are congruent.**

<fig>

<Hint>

Sol :

#### QUESTION 36

**Prove that the line segments joining the end points of two congruent arcs of a circle are wither equal or parallel**

Sol :

#### QUESTION 37

**O is the circum-centre of triangle ABC and OD is perpendicular to side BC. Prove that ∠BOD=∠A**

<fig>

<Hint>

Sol :

#### QUESTION 38

**ABC is an isosceles triangle and a line XY is drawn parallel to the side BC , which intersects equal sides at point X and Y . Prove that points B , C , Y and X are concyclic **

<fig>

<Hint>

Sol :

#### QUESTION 39

**Perpendiculars drawn from the vertices B and C of triangle ABC on the opposite sides intersect circumcircle of the ΔABC at P and Q respectively. Prove that arc ARP≅ASQ**

<fig>

<Hint>

Sol :

#### QUESTION 40

**In ΔABC , perpendiculars BE and CF are drawn from vertices B and C n opposite sides . Prove that points B , C , E and F are concyclic **

<fig>

<Hint>

Sol :

#### QUESTION 41

**AB and CD are two parallel chords of a circle whose diameter is AC . Prove that AB=CD and BC=AD**

<fig>

<Hint>

Sol :

#### QUESTION 42

**Prove that isosceles rhombus is a cyclic quadrilateral**

<fig>

<Hint>

Sol :

#### QUESTION 43

**ABCD is a quadrilateral whose vertices lie on a circle . The diagonals AC and BD intersect each other at right angle at point M . Prove that the line passing through M and bisecting a side of quadrilateral is perpendicular on the opposite side**

<fig>

<Hint>

Sol :

#### QUESTION 44

**ABCD is a cyclic quadrilateral. Bisectors of opposite angles A and C intersect the circle at points E and F respectively . Prove EF is the diameter of the circle.**

<fig>

<Hint>

Sol :

#### QUESTION 45

**Two congruent circles Intersect each other at points A and B respectively . Through A , line segment PAQ is drawn so that P and Q lie on the two circles Prove that BP=BQ**

<fig>

<Hint>

Sol :

#### QUESTION 46

**Chords AC and BD bisect each other. Prove that**

**(i) AC and BD are diameters **

**(ii) ABCD is a rectangle **

<fig>

<Hint>

Sol :

#### QUESTION 47

In a ΔABC , if bisectors of ∠A and perpendicular bisectors of BC intersect , then prove that they will intersect on the circumcircle of ΔABC

Sol :

#### QUESTION 48

**In the given figure, l is the perpendicular bisector of the side BC of ΔABC and this intersects circumcircle of ΔABC at the point M . Prove that**

<fig>

**(i) Point M lies on the centre of the circle.**

**(ii) Bisector l bisects the arc BPC**

**(iii) AM is the bisector of ∠BAC**

Sol :

#### QUESTION 49

**If diagonals of a cyclic quadrilateral intersect each other at right angles at the centre of the circle ,show that the perpendicular drawn from the centre of the circle on side being produced bisect the opposite side.**

<fig>

<Hint>

Sol :