**Exercise 16.1**

**Question 1: Fill in the blanks:**

**(i) All points lying inside/outside a circle are called ______ points/_______ points.**

**(ii) Circles having the same centre and different radii are called _____ circles.**

**(iii) A point whose distance from the centre of a circle is greater than its radius lies in _________ of the circle.**

**(iv) A continuous piece of a circle is _______ of the circle.**

**(v) The longest chord of a circle is a ____________ of the circle.**

**(vi) An arc is a __________ when its ends are the ends of a diameter.**

**(vii) Segment of a circle is a region between an arc and _______ of the circle.**

**(viii) A circle divides the plane, on which it lies, in _________ parts.**

**Solution:**

**Interior points/ Exterior**points.

**Concentric**circles.

**Exterior**of the circle.

**Arc**of the circle.

**Diameter**of the circle.

**Semi-circle**when its ends are the ends of a diameter.

**centre**of the circle.

**three**parts.

**Question 2: Write the truth value (T/F) of the following with suitable reasons:**

**(i) A circle is a plane figure.**

**(ii) Line segment joining the center to any point on the circle is a radius of the circle,**

**(iii) If a circle is divided into three equal arcs each is a major arc.**

**(iv) A circle has only finite number of equal chords.**

**(v) A chord of a circle, which is twice as long as its radius is the diameter of the circle.**

**(vi) Sector is the region between the chord and its corresponding arc.**

**(vii) The degree measure of an arc is the complement of the central angle containing the arc.**

**(viii) The degree measure of a semi-circle is 1800.**

**Solution:**

**Exercise 16.2**

**Question 1: The radius of a circle is 8 cm and the length of one of its chords is 12 cm. Find the distance of the chord from the centre.**

**Solution:**

^{2}= AC

^{2}+ OC

^{2}(Pythagoras theorem)

^{2}

^{2}= 64 – 36

^{2}= 28

**Question 2: Find the length of a chord which is at a distance of 5 cm from the centre of a circle of radius 10 cm.**

**Solution:**

^{2}= AC

^{2}+ OC

^{2}(By Pythagoras theorem)

^{2}+ 25

^{2}= 100 – 25

^{2}= 75

**Question 3: Find the length of a chord which is at a distance of 4 cm from the centre of a circle of radius 6 cm.**

**Solution:**

^{2}= AC

^{2}+ OC

^{2}(By Pythagoras theorem)

^{2}+ 16

^{2}= 36 – 16

^{2}= 20

**Question 4: Two chords AB, CD of lengths 5 cm, 11 cm respectively of a circle are parallel. If the distance between AB and CD is 3 cm, find the radius of the circle.**

**Solution:**

^{2}= OP

^{2}+ CP

^{2}(By Pythagoras theorem)

^{2}= x

^{2}+ (11/2)

^{2}________(1)

^{2}= OQ

^{2}+ AQ

^{2}(By Pythagoras theorem)

^{2}= (x-3)

^{2}+ (5/2)

^{2}________(2)

^{2}+ (5/2)

^{2}= x

^{2}+ (11/2)

^{2}(∴(a+b)

^{2}= a

^{2}+ b

^{2}+ 2ab)

^{2}+6x+9+25/4 =x

^{2}+121/4

^{2}= (5/2)

^{2}+ (11/2)

^{2}

**Question 5: Give a method to find the centre of a given circle.**

**Solution:**

**Question 6: Prove that the line joining the mid-point of a chord to the centre of the circle passes through the mid-point of the corresponding minor arc.**

**Solution:**

**Question 7: Prove that a diameter of a circle which bisects a chord of the circle also bisects the angle subtended by the chord at the centre of the circle.**

**Solution:**

**Question 8: Prove that two different circles cannot intersect each other at more than two points.**

**Solution:**

**Question 9: A line segment AB is of length 5 cm. Draw a circle of radius 4 cm passing through A and B. Can you draw a circle of radius 2 cm passing through A and B? Give reason in support of your answer.**

**Solution:**

**Question 10: An equilateral triangle of side 9 cm is inscribed in a circle. Find the radius of the circle.**

**Solution:**

**Question 11: Given an arc of a circle, complete the circle.**

**Solution:**

**Question 12: Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?**

**Solution:**

**Question 13: Suppose you are given a circle. Give a construction to find its centre.**

**Solution:**

**Question 14: Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are opposite side of its centre. If the distance between AB and CD is 6 cm, find the radius of the circle.**

**Solution:**

^{2}+ MB

^{2}= OB

^{2}(Pythagoras theorem)

^{2}+ (5/2)

^{2}= OB

^{2}

^{2}− 12x + 25/4 = OB

^{2}_______(i)

^{2}+ ND

^{2}= OD

^{2}(Pythagoras theorem)

^{2}+ (11/2)

^{2}= OD

^{2}

^{2}+ 121/4 = OD

^{2}_______ (ii)

^{2}= OD

^{2}

^{2}− 12x + 25/4 = x

^{2}+ 121/2

^{2}− 12x – x

^{2}= 121/4-25/4-36

**Question 15: The lengths of two parallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at a distance of 4 cm from the centre, what is the distance of the other chord from the centre?**

**Solution:**

^{2}= OM

^{2}+ MB

^{2}(Pythagoras theorem)

^{2}= (4)

^{2}+ (3)

^{2}

^{2}= 16 + 9

^{2}= ON

^{2}+ ND

^{2}(Pythagoras theorem)

^{2}= ON

^{2}+ (4)

^{2}

**Exercise 16.3**

**Question 1: Three girls Ishita, Isha and Nisha are playing a game by standing on a circle of radius 20 m drawn in a park. Ishita throws a ball to Isha, Isha to Nisha and Nisha to Ishita. If the distance between Ishita and Isha and between Isha and Nisha is 24 m each, what is the distance between Ishita and Nisha.**

**Solution:**

^{2 }= OA

^{2}+ AR

^{2}(Pythagoras theorem)

^{2 }= OA

^{2}+(12)

^{2 }

^{2}+144

^{2 }= 144-400

^{2}=-256

**Question 2: A circular park of radius 40 m is situated in a colony. Three boys Ankur, Amit and Anand are sitting at equal distance on its boundary each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone.**

**Solution:**

^{2}= AD

^{2}+ DC

^{2}(By Pythagoras theorem)

^{2 }= 60

^{2}+ (AC/2)

^{2}

^{2}- AC

^{2}/4 =3600

^{2}= 3600

^{2}= (3600 × 4)/3

^{2}= 4800

**Exercise 16.4**

**Question 1: In figure, O is the centre of the circle. If ∠APB = 50°, find ∠AOB and ∠OAB.**

**Solution:**

**Question 2: In figure, it is given that O is the centre of the circle and ∠AOC = 150°. Find ∠ABC.**

**Solution:**

**Question 3: In figure, O is the centre of the circle. Find ∠BAC.**

**Solution:**

**Question 4: If O is the centre of the circle, find the value of x in each of the following figures.**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Question 5: O is the circumcentre of the triangle ABC and OD is perpendicular on BC. Prove that ∠BOD = ∠A.**

**Solution:**

**Question 6: In figure, O is the centre of the circle, BO is the bisector of ∠ABC. Show that AB = AC.**

**Solution:**

**Question 7: In figure, O is the centre of the circle, then prove that ∠x = ∠y + ∠z.**

**Solution:**

**Exercise 16.5**

**Question 1: In figure, ΔABC is an equilateral triangle. Find m∠BEC.**

**Solution:**

**Question 2: In figure, Δ PQR is an isosceles triangle with PQ = PR and m∠PQR=35°. Find m∠QSR and m∠QTR.**

**Solution:**

**Question 3: In figure, O is the centre of the circle. If ∠BOD = 160°, find the values of x and y.**

**Solution:**

**Question 4: In figure, ABCD is a cyclic quadrilateral. If ∠BCD = 100° and ∠ABD = 70°, find ∠ADB.**

**Solution:**

**Question 5: If ABCD is a cyclic quadrilateral in which AD||BC (figure). Prove that ∠B = ∠C.**

**Solution:**

**Question 6: In figure, O is the centre of the circle. Find ∠CBD.**

**Solution:**

**Question 7: In figure, AB and CD are diameters of a circle with centre O. If ∠OBD = 500, find ∠AOC.**

**Solution:**

**Question 8: On a semi-circle with AB as diameter, a point C is taken, so that m(∠CAB) = 30°. Find m(∠ACB) and m(∠ABC).**

**Solution:**

**Question 9: In a cyclic quadrilateral ABCD if AB||CD and ∠B = 70° , find the remaining angles.**

**Solution:**

**Question 10: In a cyclic quadrilateral ABCD, if m ∠A = 3(m∠C). Find m∠A.**

**Solution:**

**Question 11: In figure, O is the centre of the circle ∠DAB = 50°. Calculate the values of x and y.**

**Solution:**

**Exercise VSAQs .......................**

**Question 1: In figure, two circles intersect at A and B. The centre of the smaller circle is O and it lies on the circumference of the larger circle. If ∠APB = 70°, find ∠ACB.**

**Solution:**

**Question 2: In figure, two congruent circles with centres O and O’ intersect at A and B. If ∠AO’B = 50°, then find ∠APB.**

**Solution:**

**Question 3: In figure, ABCD is a cyclic quadrilateral in which ∠BAD=75°, ∠ABD=58° and ∠ADC=77°, AC and BD intersect at P. Then, find ∠DPC.**

**Solution:**

**Question 4: In figure, if ∠AOB = 80° and ∠ABC=30°, then find ∠CAO.**

**Solution:**