**1. Using the given information, find the value of x in each of the following figures:**

**Solution:**

^{o}+ 31

^{o}+ ∠BAC = 180°

^{o}

**2. If O is the center of the circle, find the value of x in each of the following figures (using the given information):**

**Solution:**

**3. (a) In the figure (i) given below, AD || BC.If ∠ACB = 35°. Find the measurement of ∠DBC.**

**(b) In the figure (ii) given below, it is given that O is the centre of the circle and ∠AOC = 130°. Find ∠ ABC**

**Solution:**

**4. a) In the figure (i) given below, calculate the values of x and y .**

**(b) In the figure (ii) given below, O is the centre of the circle. Calculate the values of x and y.**

**Solution:**

^{o}= 180°

**5. (a) In the figure (i) given below, M,A,B,N are points on a circle having centre O. AN and MB cut at Y. If ∠NYB = 50° and ∠YNB = 20°, find ∠MAN and the reflex angle MON.**

**(b) In the figure (ii) given below, O is the centre of the circle. If ∠AOB = 140° and ∠OAC = 50°, find**

**(i) ∠ACB**

**(ii) ∠OBC**

**(iii) ∠OAB**

**(iv) ∠CBA**

**Solution:**

^{o}+ 110° + 140° + ∠OBC = 360°

**Solution:**

**8. (a) In the figure given below, P and Q are centers of two circles intersecting at B and C.ACD is a straight line. Calculate the numerical value of x.**

**(b) In the figure given below, O is the circumcenter of triangle ABC in which AC = BC. Given that ∠ACB = 56°, calculate**

**(i)∠CAB**

**(ii)∠OAC**

**Solution:**

**Exercise 15.2**

**1. If O is the center of the circle, find the value of x in each of the following figures (using the given information)**

**Solution:**

**2. (a) In the figure (i) given below, O is the center of the circle. If ∠AOC = 150°, find (i) ∠ABC (ii) ∠ADC (b) In the figure (i) given below, AC is a diameter of the given circle and ∠BCD = 75°. Calculate the size of (i) ∠ABC (ii) ∠EAF.**

**Solution:**

**3. (a) In the figure, (i) given below, if ∠DBC = 58° and BD is a diameter of the circle, calculate:**

**(i) ∠BDC (ii) ∠BEC (iii) ∠BAC**

**(b) In the figure (if) given below, AB is parallel to DC, ∠BCE = 80° and ∠BAC = 25°. Find:**

**(i) ∠CAD (ii) ∠CBD (iii) ∠ADC (2008)**

**Solution:**

**4. (a) In the figure given below, ABCD is a cyclic quadrilateral. If ∠ADC = 80° and ∠ACD = 52°, find the values of ∠ABC and ∠CBD.**

**(b) In the figure given below, O is the center of the circle. ∠AOE =150°,∠DAO = 51°. Calculate the sizes of ∠BEC and ∠EBC.**

**Solution:**

**5. (a) In the figure (i) given below, ABCD is a parallelogram. A circle passes through A and D and cuts AB at E and DC at F. Given that ∠BEF = 80°, find ∠ABC.**

**(b) In the figure (ii) given below, ABCD is a cyclic trapezium in which AD is parallel to BC and ∠B = 70°, find:**

**(i)∠BAD (ii) DBCD.**

**Solution:**

**6. (a) In the figure given below, O is the center of the circle. If ∠BAD = 30°, find the values of p, q and r.**

**(a) In the figure given below, two circles intersect at points P and Q. If ∠A = 80° and ∠D = 84°, calculate**

**(i) ∠QBC**

**(ii) ∠BCP**

**Solution:**

**7. (a) In the figure given below, PQ is a diameter. Chord SR is parallel to PQ.Given ∠PQR = 58°, calculate (i) ∠RPQ (ii) ∠STP**

**(T is a point on the minor arc SP)**

**(b) In the figure given below, if ∠ACE = 43° and ∠CAF = 62°, find the values of a,b and c .**

**Solution:**

**8. (a) In the figure (i) given below, AB is a diameter of the circle. If ∠ADC = 120°, find ∠CAB.**

**(b) In the figure (ii) given below, sides AB and DC of a cyclic quadrilateral ABCD are produced to meet at E, the sides AD and BC are produced to meet at F. If x∶ y∶ z = 3∶ 4∶ 5, find the values of x,y and z.**

**Solution:**

**Exercise 15.3**

**1. Find the length of the tangent drawn to a circle of radius 3cm, from a point distnt 5cm from the center.**

**Solution:**

^{2}= OT

^{2}+ PT

^{2}

^{2}= (3)2 + PT

^{2}

^{2}= (5)2 – (3)2 = 25 – 9 = 16 = (4)

^{2}

**2. A point P is at a distance 13 cm from the center C of a circle and PT is a tangent to the given circle. If PT = 12 cm, find the radius of the circle.**

**Solution:**

^{2}= CT

^{2}+ PT

^{2}[using Pythagoras axiom]

^{2}= (CT)

^{2}+ (12)

^{2}

^{2}+ 144

^{2}= 169 -144 =25 = (5)

^{2}

**3. The tangent to a circle of radius 6 cm from an external point P, is of length 8 cm. Calculate the distance of P from the nearest point of the circle.**

^{2}= OA

^{2}+ AP

^{2}

^{2}+ (8)

^{2}

^{2}

**4. Two concentric circles are of the radii 13 cm and 5 cm. Find the length of the chord of the outer circle which touches the inner circle.**

**We have given that:**

**Solution:**

^{2}= OP

^{2}+ PB

^{2}

^{2}= 5

^{2}+ PB

^{2}

^{2}

^{2}= 169 – 25

**5. Two circles of radii 5 cm and 2-8 cm touch each other. Find the distance between their centers if they touch :**

**(i) externally**

**(ii) internally.**

**Solution:**

**6. (a) In figure (i) given below, triangle ABC is circumscribed, find x.**

**(b) In figure (ii) given below, quadrilateral ABCD is circumscribed, find x.**

**Solution:**

**7. (a) In figure (i) given below, quadrilateral ABCD is circumscribed; find the perimeter of quadrilateral ABCD.**

**(b) In figure (ii) given below, quadrilateral ABCD is circumscribed and AD ⊥ DC ; find x if radius of incircle is 10 cm.**

**Solution:**

**8. (a) In the figure (i) given below, O is the center of the circle and AB is a tangent at B. If AB = 15 cm and AC = 7.5 cm, find the radius of the circle.**

**(b) In the figure (ii) given below, from an external point P, tangents PA and PB are drawn to a circle. CE is a tangent to the circle at D. If AP = 15 cm, find the perimeter of the triangle PEC.**

**Solution:**

^{2}= OA

^{2}– AB

^{2}

^{2}= (r + 7.5)

^{2}– 15

^{2}

^{2}= r

^{2}+ 56.25 + 15r – 225

**Solution:**

^{2}= 400 + 169

**10. Three circles of radii 2 cm,3 cm and 4 cm touch each other externally. Find the perimeter of the triangle obtained on joining the centers of these circles.**

**Solution:**

**Chapter Test**

**1. (a) In the figure (i) given below, triangle ABC is equilateral. Find ∠BDC and ∠BEC. (b) In the figure (ii) given below, AB is a diameter of a circle with center O.OD is perpendicular to AB and C is a point on the arc DB. Find ∠BAD and ∠ACD**

**Solution:**

**Solution:**

**3. a) In the figure (i) given below, O is the centre of the circle. Prove that ∠AOC = 2 (∠ACB + ∠BAC). (b) In the figure (ii) given below, O is the centre of the circle. Prove that x + y = z**

**Solution :**