## Selina Concise Mathematics Class 6 ICSE Solutions – The Circle

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**IMPORTANT POINTS**

**1.** A circle is a round enclosed figure, whose mid-point is called its centre.

**2.** The line segment joining the centre to any point on the circle is called a radius. A centre has infinite radii and all radii of a circle are equal.

**3.** A line segment which contains the centre of the circle and whose ends points lie on the circle is called diameter of the circle. Diameters of a circle are also equal.

**4. Parts of a circle:** A circle has three parts (i) Interior (ii) Exterior and (iii) Circle itself.

**5. Concentric circles:** Two or more circles having the same centre but different radii are called concentric circles.

**6. Chord of a circle:** A line which divides the circle into two parts is called chord of the circle. Diameter is the longest chord of the circle.

**7. Segment of a circle:** When a chord divides the circle into two unequal parts, the bigger part is called the major segment and smaller part is called the minor segment.

**8. Arc of a circle:** A part of circumference of a circle is called an arc of the circle. Arc greater than half circle is called the major arc and less than half circle is called the minor arc.

**9. Sector of a circle:** A diameter divides the circle into two equal parts and each part is called a semicircle. Sector greater than a semi-circle is called the major sector and less than semi-circle is called the minor sector of the circle.

**EXERCISE 29 (A)**

**Question 1.**

**Use the figure given below to fill in the blanks :**

**(i) R is the …… of the circle.**

**(ii) Diameter of a circle is …… .**

**(iii) Tangent to a circle is … .**

**(iv) EF is a …… of the circle.**

**(v) …… is a chord of the circle.**

**(vi) Diameter = 2 x …..**

**(vii) ……. is a radius of the circle.**

**(viii) If the length of RS is 5 cm, the length of PQ = ……**

**(ix) If PQ is 8 cm long, the length of RS =…..**

**(x) AB is a ….. of the circle**

**Solution:**

**(i)** center

**(ii)** PQ

**(iii)**AB

**(iv)** secant

**(v)** CD

**(vi)** radius

**(vii)** RS

**(viii)** 10 cm

**(ix)** 4 cm

**(x)** tangent.

**Question 2.**

**Draw a circle of radius 4.2 cm. Mark its centre as O. Take a point A on the circumference of the circle. Join AO and extend it till it meets point B on the circumference of the circle,
(i) Measure the length of AB.**

**(ii) Assign a special name to AB.**

**Solution:**

**(i) By measurement AB = 8.4 cm.**

**(ii) ∴ AB is the diameter of the circle.**

**Question 3.**

**Draw circle with diameter :**

**(i) 6 cm**

**(ii) 8.4 cm.**

**In each case, measure the length of the radius of the circle drawn.**

**Solution:**

**Question 4.**

**Draw a circle of radius 6 cm. In the circle, draw a chord AB = 6 cm.**

**(i) If O is the centre of the circle, join OA and OB.**

**(ii) Assign a special name to ∆AOB**

**(iii) Write the measure of angle AOB.**

**Solution:**

**Question 5.**

**Draw a circle of radius 4.8 cm and mark its centre as P.**

**(i) Draw radii PA and PB such that ∠APB = 45°.**

**(ii) Shade the major sector of the circle**

**Solution:**

PA is the radius of circle. i.c., PA = 4.8 cm.

**(i)** ∠APB = 45° in which P is the centre of the circle and PA and PB are radii of circle.

**(ii)** Major sector of circle is shaded in the figure.

**Question 6.**

**Draw a circle of radius 3.6 cm. In the circle, draw a chord AB = 5 cm. Now shade the minor segment of the circle.**

**Solution:**

Shaded portion of circle is the minor segment of the circle.

**Question 7.**

**Mark two points A and B ,4cm a part, Draw a circle passing through B and with A as a center**

**Solution:**

In the figure, A is the centre of the circle and AB = 4 cm [radius of circle]

**Question 8.**

**Draw a line AB = 8.4 cm. Now draw a circle with AB as diameter. Mark a point C on the circumference of the circle. Measure angle ACB.**

**Solution:**

By measurement ∠ACB =90

**EXERCISE 29 (B)**

**Question 1.**

**Construct a triangle ABC with AB = 4.2 cm, BC = 6 cm and AC = 5cm. Construct the circumcircle of the triangle drawn.**

**Solution:**

**Steps of Construction :**

**(i)** Draw ∆ABC in which AB = 4.2 cm. BC = 6 cm. and AC = 5 cm.

**(ii)** Draw the perpendicular bisectors of any two sides of the triangle. Let these intersect at O.

**(iii)** Taking O as centre and OA or OB or OC as radius draw a circle.

This circle will pass through vertices A, B and C.

**Question 2.**

**Construct a triangle PQR with QR = 5.5 cm, ∠Q = 60° and angle R = 45°.**

**Construct the circumcircle cif the triangle PQR.**

**Solution:**

**Steps of Construction :**

**(i)** Draw a ∆PQR in which QR = 5.5 cm, ∠Q = 60° and ∠R = 45°.

**(ii)** Draw the arc bisector of PQ and PR which intersect at O.

**(iii)** Taking O as centre and radius OP or OQ or OR draw a circle.

This circle will pass through vertices P, Q and R.

**Question 3.**

**Construct a triangle ABC with AB = 5 cm, ∠B = 60° and BC = 6. 4 cm.**

**Draw the incircle of the triangle ABC. Sol. Steps of Construction :**

**Solution:**

**Steps of Construction:**

**(i)** Draw a line AB = 5 cm.

**(ii)**B as a centre draw an angle with the help of compass ∠B = 60°. Cut the line with an arc BC = 6 4 cm.

**(iii)** Join AC.

**(iv)** Now, from A and B cut the bisector of ∠A and ∠B, which intersect each other at point D.

**(v)** With D as a centre draw an incircle which touches all the three sides of AABC.

**Question 4.**

**Construct a triangle XYZ in which XY = YZ= 4.5 cm and ZX = 5.4 cm. Draw the circumcircle of the triangle and measure its circumradius.**

**Solution:**

**Steps of Construction :**

**(i)** Draw a triangle XYZ in which XY = YZ = 4.5 cm and ZX = 5.4 cm.

**(ii)** Draw the bisectors of XZ and YZ which meet at O.

**(iii)** With O as centre and radius OX or OY or OZ draw a circle.

This circle will pass through X, Y and Z.

**Question 5.**

**Construct a triangle PQR in which, PQ = QR = RP = 5.7 cm. Draw the incircle of the triangle and measure its radius.**

**Solution:**

**Steps of Construction :**

**(i)** Draw an equilateral ∆ RPQ in which PQ = QR = RP = 5.7 cm each.

**(ii)** From P and Q cut the bisector of ∠P and ∠Q, which intersect each other at point O.

**(iii)** With P as a centre draw an incircle which touches all the three sides of ∆RPQ.

**REVISION EXERCISE**

**Question 1.**

**The centre of a circle is at point O and its radius is 8 cm. State the position of a point P (point P may lie inside the circle, on the circumference of the circle, or outside the circle), when :**

**(a) OP = 10.6 cm**

**(b) OP = 6.8 cm**

**(c) OP = 8 cm**

**Solution:**

**(a)** Draw circle each of radius 8 cm. With centre O

In figure (i) draw OP = 10.6 cm

We see that point P lies outside the circle as OP > radius of the circle

**(b)** In figure (ii) OP = 6.8 cm. We see that P lies inside the circle as OP < radius of the circle.

**(c)** In figure, OP = 8 cm. We see that P lies on the circle as OP = radius of the circle.

**Question 2.**

**The diameter of a circle is 12.6 cm. State, the length of its radius.**

**Solution:**

Diameter of the circle = 12.6 cm

∴Radius = diameter = x 12.6 cm

= 6.3 cm

**Question 3.**

**Can the length of a chord of a circle be greater than its diameter ? Explain.**

**Solution:**

No, the length of chord cannot be greater than the diameter of the cirlce as the diameter of a circle is the greatest chord of that circle.

**Question 4.**

**Draw a circle of diameter 7 cm. Draw two radii of this circle such that the angle between these radii is 90°. Shade the minor sector obtained. Write a special name for this sector.**

**Solution:**

Draw a circle with diameter = 7 cm

OA and OB are the radii of the circle such that ∠AOB = 90°

Now shade the minor sector AOB This is the quadrant of the circle

**Question 5.**

**State, which of following statements are true and which are false :**

**(i) If the end points A and B of the line segment lie on the circumference of a circle, AB is a diameter.**

**(ii) The longest chord of a circle is its diameter.**

**(iii) Every diameter bisects a circle and each part of the circle so obtained is a semi-circle.**

**(iv) The diameters of a circle always pass through the same point in the circle.**

**Solution:**

(i) False, as AB may be diameter or may not be, it can be chord.

(ii) True, diameter of a circle is the longest chord.

(iii) True.

(iv) True, all the diameter of a circle pass through the same point i.e., centre, of the circle.

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