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Neokaal Study

Circles

Practice Worksheet

Tangent lengths, tangent angles, and theorem-based reasoning with circles.

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  1. From an external point T, tangent TA touches a circle with centre O. If $OT=14$ cm and $\angle OTA=30^\circ$, find TA.

    1. (A)$7\sqrt{2}$ cm
    2. (B)$7\sqrt{3}$ cm
    3. (C)$14\sqrt{3}$ cm
    4. (D)$7$ cm

  2. Two tangents from P to a circle of radius 7 cm make an angle of 90°. Find the length of each tangent.

    1. (A)7 cm
    2. (B)14 cm
    3. (C)$7\sqrt{2}$ cm
    4. (D)$7/2$ cm

  3. A point P is 39 cm from the centre O of a circle of radius 36 cm. Tangents PA and PB touch the circle. Find the area of quadrilateral PAOB.

    1. (A)270 cm²
    2. (B)585 cm²
    3. (C)540 cm²
    4. (D)1404 cm²

  4. Tangents at A and B to a circle meet at P. If $\angle AOB=145^\circ$, find $\angle APB$.

    1. (A)18°
    2. (B)35°
    3. (C)72°
    4. (D)145°

  5. Tangents PA and PB meet at P, and $\angle APB=100^\circ$. Find $\angle OAB$, where O is the centre.

    1. (A)80°
    2. (B)50°
    3. (C)30°
    4. (D)40°

  6. Assess this claim as True or False: chord AB makes a 60° central angle, so the two tangents through A and B must also enclose 60°.

  7. Assess this claim as True or False: no matter where an outside point is chosen, its tangent length exceeds the radius of the circle.

  8. Assess this claim as True or False: when PT touches a circle centred at O, the segment PT is shorter than the centre-distance OP.

  9. Assess this claim as True or False: a finite point outside a circle can have two tangent rays whose included angle is 0°.

  10. Assess this claim as True or False: tangents from P enclose 90° around a circle of radius 9 cm, which makes P exactly $9\sqrt{2}$ cm from the centre.

  11. Assess this claim as True or False: fix A on segment PQ and allow circles tangent to PQ at A; every possible centre lies on the perpendicular bisector of PQ.

  12. Assess this claim as True or False: as a circle varies while continuing to pass through fixed points P and Q, its centre remains on the perpendicular bisector of PQ.

  13. A circle has centre O. From an outside point P, the contact points of the two tangents are Q and R. Show that the four points Q, O, R, and P lie on one circle.

  14. An outside point B is joined tangentially to a circle at C and D, with each tangent measuring 7 cm. The angle between them is $120^\circ$. Establish that the distance from B to the centre O is twice BC.

  15. Two non-parallel lines are both tangent to the same circle. Show that the circle's centre must be on one of the bisectors of the angle formed by the lines.

  16. From P, two rays are direct common tangents to a pair of unequal circles. Along the rays, the contact points occur in the orders P-A-B and P-C-D. Demonstrate that the portions between the circles, AB and CD, have the same length.

  17. For two congruent circles, mark the contact-to-contact portions of their direct common tangents as AB and CD. Show that these two portions are equal.

  18. Two common tangents cross at E between a pair of circles. Their complete contact-to-contact segments are AB and CD. Use tangent lengths from E to show that these segments are equal.

Neokaal Study

Answer Key

  1. $7\sqrt{3}$ cm

    1. Since OA is a radius to the point of contact, $OA\perp TA$.
    2. In right triangle OTA, $\cos 30^\circ=TA/14$.
    3. Therefore $TA=14\times\frac{\sqrt{3}}{2}=7\sqrt{3}$ cm.

  2. 7 cm

    1. The line joining P to the centre bisects the angle between the equal tangents.
    2. Use one right triangle: its angle at P is $90/2=45^\circ$, with radius opposite and tangent adjacent.
    3. Thus $\tan 45^\circ=\text{radius}/\text{tangent}$, giving tangent length 7 cm.

  3. 540 cm²

    1. A radius is perpendicular to the tangent at its point of contact.
    2. In right triangle OAP, $PA=\sqrt{39^2-36^2}=15$ cm.
    3. The quadrilateral consists of two right triangles, so its area is $2\times\frac12\times 36\times 15=540$ cm².

  4. 35°

    1. The radii OA and OB are perpendicular to the tangents PA and PB.
    2. Thus quadrilateral OAPB has two right angles.
    3. So $\angle APB=180^\circ-145^\circ=35^\circ$.

  5. 40°

    1. The equal tangents PA and PB make the configuration symmetric about OP.
    2. The angle between radii OA and OB is $180^\circ-100^\circ=80^\circ$.
    3. Triangle AOB is isosceles, so each base angle is $40^\circ$.

  6. False

    1. Each radius is perpendicular to its tangent.
    2. The central angle and the angle between the tangents are supplementary.
    3. The tangent angle is $180^\circ-60^\circ=120^\circ$, not 60°.

  7. False

    1. The tangent length depends on how far the external point is from the circle.
    2. A valid right-triangle configuration can have radius 12 units and tangent length 5 units.
    3. This counterexample has tangent length less than the radius, so the word ‘always’ makes the claim false.

  8. True

    1. Radius OT is perpendicular to tangent PT.
    2. Therefore triangle OPT is right-angled at T, making OP its hypotenuse.
    3. The hypotenuse is longer than either leg, so $PT<OP$.

  9. False

    1. The two tangents touch the circle at distinct points.
    2. They are distinct intersecting lines through the external point.
    3. Their smaller angle is positive; 0° would mean that the two tangent rays coincide.

  10. True

    1. The centre line bisects the 90° angle between the equal tangents.
    2. One half is a 45°–45°–90° right triangle with the radius as a leg.
    3. Its hypotenuse is therefore $9\sqrt{2}$ cm, so the claim is true.

  11. False

    1. The radius to A must be perpendicular to line PQ.
    2. Therefore the centres lie on the line through A perpendicular to PQ.
    3. That line is the perpendicular bisector of PQ only when A is the midpoint, so the general claim is false.

  12. True

    1. P and Q are points on each circle, so OP and OQ are radii of that circle.
    2. Hence every possible centre O is equidistant from P and Q.
    3. The locus of points equidistant from P and Q is their perpendicular bisector.

  13. Proved

    1. A radius is perpendicular to the tangent at its point of contact, so $\angle OQP=90^\circ$.
    2. Similarly, $\angle ORP=90^\circ$.
    3. These opposite angles sum to 180°, so QORP is a cyclic quadrilateral.

  14. Proved

    1. Triangles OCB and ODB are congruent, so BO bisects the 120° angle.
    2. Thus $\angle CBO=60^\circ$, while OC is perpendicular to BC.
    3. In right triangle BCO, $\cos60^\circ=BC/BO=1/2$, hence $BO=2BC$.

  15. Proved

    1. Draw radii from the centre O to the two points of contact.
    2. Both radii are perpendicular to their respective lines and have equal length.
    3. Therefore O is equidistant from the two lines; the locus of such points is an angle bisector.

  16. Proved

    1. Tangents from P to the first circle are equal, so PA = PC.
    2. Tangents from P to the second circle are equal, so PB = PD.
    3. Since AB = PA - PB and CD = PC - PD, subtraction gives AB = CD.

  17. Proved

    1. The radii to A and B are equal and perpendicular to their common tangent.
    2. Together with the line of centres, they form a rectangle, so AB equals the distance between the centres.
    3. The same argument gives CD equal to that centre distance; hence AB = CD.

  18. Proved

    1. From E, the tangent segments to the first circle are equal: EA = EC.
    2. The tangent segments to the second circle are also equal: EB = ED.
    3. Since AB = AE + EB and CD = CE + ED, the two common-tangent segments are equal.