ICSE Board: Suggested Books     ICSE Board:  Foundation Mathematics
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Q.1. Fill in the blanks

1. A line segment joining any point on the circle to its center is called a radius  of the circle.
2. All the radii of a circle are equal.
3. A line segment having its end points on a circle is called chord  of a circle.
4. A chord that passes through the center of the circle is called a diameter  of the circle.
5. Diameter of a circle is twice  its radius.
6. A diameter is the largest  chord of the circle.
7. The interior of a circle together with the circle is called the area of the circle.
8. A chord of a circle divides the whole circular region into two parts, each called a  segment.
9. Half of a circle is called a semicircle.
10. A segment of a circle containing the center is called the major segment of the circle.
11. The mid point of the diameter of a circle is the center  of the circle.
12. The perimeter of the circle is called its circumference.

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Q.2. State which of the following statements are true or false:

1. Diameter of a circle is a part of a semi-circle of a circle : True
2. Two semi-circles of a circle together make the whole circle: True
3. Two semi-circular regions of a circle together make the whole circular region:  True
4. An infinite number of chords may be drawn in a circle: True
5. A line can meet a circle at the most at two points: True
6. An infinite number of diameters can be drawn in a circle: True
7. A circle has an infinite number of radii: True
8. A circle consists of an infinite number of points: True
9. Center of a circle lies on a circle: True

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Q.3. From an external point P, 29 cm away from the center of a circle, a tangent PT of length 21 cm is drawn. Find the radius of the circle.

Radius $= \sqrt{(29^2-21^2 )}= 20 cm$

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Q.4. Two tangents  $PM \ and\ PN$ are drawn from an exterior point p to a circle with center O. Prove that: $\angle OPM \cong \angle OPN$

In  $\Delta PMO \ and\ \Delta PNO,$

PO is common, OM=ON (radius of the circle)  and

$PN=PM$ (Tangents to a circle from one point  circle are equal.)

Hence   $\Delta PMO \cong \Delta PNO$

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Q.5. In the given figure,  $\Delta ABC$is inscribed in a circle with center O. If  $\angle ACB=40^{\circ}$, find angle

$\angle ABC+ \angle BCA+ \angle BAC=180^{\circ}$

$\angle BAC=90$  (angle in a semicircle is a right angle).

$\Rightarrow \angle ABC=180-90-40=50^{\circ}$

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Q.6. In the given figure, O is the centre of a circle.  $\Delta ABC$ is inscribed in this circle. If  $AB = AC, \ find\ \angle ABC \ and\ \angle ACB$.

$\angle BAC=90^{\circ}$  (angle in a semicircle is a right angle).

$AB=AC$

$\Rightarrow \angle CBA= \angle CBA=x^{\circ}$

$\Rightarrow 2x+90=180$

$\Rightarrow x=45^{\circ}$

$\therefore \angle CBA= \angle CBA=45^{\circ}$

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Q.7. In the given figure, O is the centre of a circle. If  $\angle ABC = 54^{\circ}$ , find $\angle ACB$. Also, if $\angle BCD = 43^{\circ}$  , find $\angle CBD$.

$\angle CBA+ \angle BAC+ \angle ACB=180$

$\angle BAC=90^{\circ}$  (angle in a semicircle is a right angle).

$\Rightarrow \angle ACB=180-90-54=36^{\circ}$

Similarly

$\angle CBD+ \angle CDB+ \angle BCD=180^{\circ}$

$\angle BDC=90^{\circ}$   (angle in a semicircle is a right angle).

$\Rightarrow \angle CBD=180-90-43=47^{\circ}$

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Q.8. In the given figure, $\Delta ABC$ is inscribed in a circle with center O. If $\angle ABC=(3x - 7)^{\circ} \ and\ \angle ACB=(x -7)^{\circ}$ , find the value of  $x$.

$\angle CBA+ \angle BAC+ \angle ACB=180$

$\angle BAC=90^{\circ}$ (angle in a semicircle is a right angle).

$\Rightarrow (3x-7)+90+(x+13)=180 \Rightarrow x=21^{\circ}$

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Q.9. In the adjoining figure, PRT is a tangent to the circle with center O. QR is a diameter of the circle. If  $\angle QPR = 53^{\circ} \ and\ \angle PQR = x^{\circ}$ , then find the value of x.

$\angle RPQ+ \angle PQR+ \angle QRP=180$

$\angle QRP=90^{\circ}$   (tangent is perpendicular to the line drawn from the center to the point of contact)

$\Rightarrow \angle PQR=x=37^{\circ}$

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Q.10. In the adjoining figure, PRT is a tangent to the circle with centre O. OR is the radius of the circle at the point of contact. P, O are joined and produced to the point Q on the circle. If  $\angle RPO = 28^{\circ} , \angle POR =x^{\circ} \ and\ \angle ORQ =y^{\circ}$ , then find the values of $x and y$.

$\angle PRO=90$  (tangent is perpendicular to the line drawn from the center to the point of contact)

$\Rightarrow 90+x+28=180\Rightarrow x=62^{\circ}$

Also  $2 \angle RQO= \angle ROP$

$\Rightarrow \angle ORQ+\angle RQO+ \angle QOR=180$

$\Rightarrow 31+118+y=180$

$\Rightarrow y=31^{\circ}$

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Q.11. In the adjoining figure, PT is a tangent to the circle with center O, QT is a diameter of the circle. If  $PT = QT \ and\ \angle QPT = x^{\circ}$ , then find the value of $x$.

Given  $PT = QT$

$\Rightarrow \angle QPT= \angle PQT = x$

$\Rightarrow 2x+90=180 \Rightarrow x=45^{\circ}$

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Q.12. In the adjoining figure, PX and PY are tangents drawn from an exterior point P to a circle with centre O and radius 8 cm. If PX = 15 cm, OP = a cm, PY = b cm,  $\angle POX = 56^{\circ} \ and\ \angle OPY = x^{\circ}$  then find the value of $a,\ b\ and\ x$.

$a= OP= \sqrt{(8^2+15^2 )}= 17 cm$

$b= PY= \sqrt{(17^2-8^2 )}= 15 cm$

$\angle POX = \angle POY=56$

$\Rightarrow 56+90+x=180$

$\Rightarrow x=34^{\circ}$

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Q.13. In the adjoining figure, AB is the diameter of the circle with centre O. If  $\angle ABM = 124^{\circ} \ and\ CAB = x^{\circ}$, then find the value of $x$.

$180- 124+90+x=180 \Rightarrow$

$x=34^{\circ}$

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Q.14. In the adjoining figure, PQ is a diameter of a circle with center O.  $\Delta PQR$ is an isosceles triangle with RP=RQ.  PQ is produced to a point S such that RQ = QS. If  $\angle QPR = x \ and\ \angle QSR =y$, then find the values of $x \ and\ y$.

In  $\Delta PQR, RP=RQ \Rightarrow \angle RPQ= \angle RQP=x$
And we know  $\angle PRQ=90 \Rightarrow x=45^{\circ}$
$\angle RQS=180-45=135^{\circ}$
Given  $RQ=QS \Rightarrow \angle QRS=y$
$\Rightarrow In \Delta PRS, 45+y+y+90=180 \Rightarrow y=22 \frac{1}{2}^{\circ}$