Selina solutions for Concise Mathematics [English] Class 10 ICSE chapter 18 - Tangents and Intersecting Chords [Latest edition]

In the given figure, PA, PB and QR are tangents to a circle. If perimeter of the △PQR = 18 cm, the length of tangent PA is:

In the given figure, APB is tangent to the inner circle and also a chord of outer circle. Both the circles are concentric. If OA = 10 cm and OP = 6 cm, the length of AB is:

A, B and C are three circles which touch each other as shown. Using the information, given in the diagram, we find the length AB as:

BC is a tangent to the circle with centre O. OD is radius of the circle. If ∠DOC = 100°, ∠B is equal to:

PA and PB are tangents to a circle with centre O. If angle BPA = 70°, the angle ACB is:

In the given circle with centre O, PA and PB are tangents and ∠OAB = 28°, then ∠APB is:

In the above figure, if ∠P = 50°, then reflex angle AOB is ______.

In the given figure, O is the center of the circle and AB is a tangent at B. If AB = 15 cm and AC = 7.5 cm, calculate the radius of the circle.

If the sides of a quadrilateral ABCD touch a circle, prove that:

AB + CD = BC + AD.

From the given figure, prove that : AP + BQ + CR = BP + CQ + AR. 

Also show that : AP + BQ + CR = `1/2` × Perimeter of ΔABC.

In the following figure; If AB = AC then prove that BQ = CQ.

Radii of two circles are 6.3 cm and 3.6 cm. State the distance between their centres if:

1. they touch each other externally,
2. they touch each other internally.

In the given figure, two circles touch each other externally at point P. AB is the direct common tangent of these circles. Prove that :

1. tangent at point P bisects AB,
2. angles APB = 90°.

Tangents AP and AQ are drawn to a circle, with centre O, from an exterior point A. Prove that : ∠PAQ = 2∠OPQ

ABC is a right angles triangle with AB = 12 cm and AC = 13 cm. A circle, with centre O, has been inscribed inside the triangle.

Calculate the value of x, the radius of the inscribed circle.

In the given figure, PT touches the circle with centre O at point R. Diameter SQ is produced to meet the tangent TR at P. Given ∠SPR = x° and ∠QRP = y°;

Prove that:

1. ∠ORS = y°
2. write an expression connecting x and y.

PT is a tangent to the circle at T. If ∠ABC = 70° and ∠ACB = 50°; calculate:

1. ∠CBT
2. ∠BAT
3. ∠APT

In the given figure, O is the centre of the circumcircle ABC. Tangents at A and C intersect at P. Given angle AOB = 140° and angle APC = 80°; find the angle BAC.

In the given figure, PQ is a tangent to the circle at A. AB and AD are bisectors of ∠CAQ and ∠PAC. If ∠BAQ = 30°, prove that : BD is diameter of the circle.

Chords AB and CD of a circle intersect each other at point O such that OA:OC= 4 : 7. Then OB : OD is equal to:

If angle PAC : angle PCA = 5: 4 ; angle P is ______.

AC is a tangent to the given circle which touches the circle at point B . If angle EBC = 45°; angle EDB is equal to:

In the given circle, PA is tangent and PBC is secant, PA = 8 cm and PB = 4 cm. The length of BC is:

In the given figure, O is the centre of the circle, PA is tangent and PBC is secant. If angle ABC = 60°; angle P is:

In the given figure, diameter AB and chord CD of a circle meet at P. PT is a tangent to the circle at T. CD = 7.8 cm, PD = 5 cm, PB = 4 cm. Find:

1. AB.
2. the length of tangent PT.

In the following figure, PQ is the tangent to the circle at A, DB is the diameter and O is the centre of the circle. If ∠ADB = 30° and ∠CBD = 60°, calculate:

1. ∠QAB, 
2. ∠PAD, 
3. ∠CDB.

If PQ is a tangent to the circle at R; calculate:

1. ∠PRS,
2. ∠ROT.

Given O is the centre of the circle and angle TRQ = 30°.

Two circle with centres O and O' are drawn to intersect each other at points A and B. Centre O of one circle lies on the circumference of the other circle and CD is drawn tangent to the circle with centre O' at A. Prove that OA bisects angle BAC.

In the figure, ABCD is a cyclic quadrilateral with BC = CD. TC is tangent to the circle at point C and DC is produced to point G. If ∠BCG = 108° and O is the centre of the circle, find :

1. angle BCT
2. angle DOC

In the figure; PA is a tangent to the circle, PBC is secant and AD bisects angle BAC. Show that triangle PAD is an isosceles triangle. Also, show that:

`∠CAD = 1/2 (∠PBA - ∠PAB)`

Two circles intersect each other at points A and B. Their common tangent touches the circles at points P and Q as shown in the figure. Show that the angles PAQ and PBQ are supplementary.

In the figure, chords AE and BC intersect each other at point D. If ∠CDE = 90°, AB = 5 cm, BD = 4 cm and CD = 9 cm; find DE.

In the figure, chords AE and BC intersect each other at point D. If AD = BD, show that AE = BC.

In the adjoining figure, O is the centre of the circle and AB is a tangent to it at point B. ∠BDC = 65°. Find ∠BAO.

AP is a tangent to the given circle. If AB = 8 cm and BC = 10 cm, then AP is:

In the given figure, O is centre of the circle and PQ is a tangent. If angle OAB = x; the measure of angle ABP, in terms of x, is:

In the given figure, AB is tangent to the circle with centre O. If OCB is a straight line segment, the angle BAC is:

In the given figure O is centre, PQ is tangent at point A. BD is diameter and ∠AOD = 84° then angle QAD is:

Two mutually perpendicular tangents are drawn to a circle with radius R√2 units. The shortest distance between the two points of contact is:

Three circles with centres A, B and C and radii 5 cm, 2 cm and 6 cm respectively and touch each other externally.

Assertion (A): To find the perimeter of the triangle ABC, add the radii of given three circles.

Reason (R): The required perimeter is the product of the sum of the radii and 2.

AB is diameter of the circle, PA is tangent and ∠AOC = 60°.

Assertion (A): x + 30° = 90°.

Reason (R): PA is tangent
⇒ ∠BAP = 90°
∴ x + 30° = 90°

Chords AD and BC when produced meet at exterior point P.

Assertion (A): PD × AD = PC × BC

Reason (R): In triangles PAB and PCD,

∠PAB = ∠PCD ⇒ △PAB ∼ △PCD

⇒ `(PD)/(PB) = (PC)/(PA)`

Two circles touch each other externally at point P. OA and OB are tangents to the two circles (as shown) and OA = 10.

Statement (1): OB = 10 cm.

Statement (2): On joining O and P, tangent OP = tangent OA and tangent OP = tangent OB.

O is centre of the circle, PB and PC are tangents and ∠BPC = 50°.

Statement (1): ∠BAC = ∠P = 50°

Statement (2): ∠BOC + 50° = 180°

⇒ ∠BOC = 130°

∴ ∠BAC = 65°

In the given figure, C and D are points on the semi-circle described on AB as diameter. Given angle BAD = 70° and angle DBC = 30°, calculate angle BDC.

In cyclic quadrilateral ABCD, ∠A = 3∠C and ∠D = 5∠B. Find the measure of each angle of the quadrilateral.

Show that the circle drawn on any one of the equal sides of an isosceles triangle as diameter bisects the base.

Bisectors of vertex angles A, B, and C of a triangle ABC intersect its circumcircle at the points D, E and F respectively. Prove that angle EDF  = 90° – `1/2` ∠A.

In the figure, AB is the chord of a circle with centre O and DOC is a line segment such that BC = DO. If ∠C = 20°, find angle AOD.

P is the mid-point of an arc APB of a circle. Prove that the tangent drawn at P will be parallel to the chord AB.

In the given figure, ABCD is a cyclic quadrilateral, PQ is tangent to the circle at point C and BD is its diameter. If ∠DCQ = 40° and ∠ABD = 60°, find;

1. ∠DBC
2. ∠BCP
3. ∠ADB

The given figure shows a circle with centre O and BCD is tangent to it at C. Show that : ∠ACD + ∠BAC = 90°.

ABC is a right triangle with angle B = 90°, A circle with BC as diameter meets hypotenuse AC at point D. Prove that: AC × AD = AB²

ABC is a right triangle with angle B = 90º. A circle with BC as diameter meets by hypotenuse AC at point D. Prove that: BD² = AD × DC.

In the given figure, AC = AE. Show that:

1. CP = EP
2. BP = DP

In the given figure, O is the centre of the circle. Tangents at A and B meet at C. If ∠ACO = 30°, find:

1. ∠BCO 
2. ∠AOB
3. ∠APB

The given figure shows a semi-circle with centre O and diameter PQ. If PA = AB and ∠BCQ =140°; find measures of angles PAB and AQB. Also, show that AO is parallel to BQ.

The given figure shows a circle with centre O such that chord RS is parallel to chord QT, angle PRT = 20° and angle POQ = 100°. Calculate:

1. angle QTR
2. angle QRP
3. angle QRS
4. angle STR

In the given figure, XY is the diameter of the circle and PQ is a tangent to the circle at Y.

If ∠AXB = 50° and ∠ABX = 70°, find ∠BAY and ∠APY.

In the given figure, QAP is the tangent at point A and PBD is a straight line.

If ∠ACB = 36° and ∠APB = 42°, find:

1. ∠BAP
2. ∠ABD
3. ∠QAD
4. ∠BCD

In the given figure, AB is the diameter. The tangent at C meets AB produced at Q. If ∠CAB = 34°, find:

1. ∠CBA
2. ∠CQB

In the given figure, O is the centre of the circle. The tangents at B and D intersect each other at point P. If AB is parallel to CD and ∠ABC = 55°, find:

1. ∠BOD
2. ∠BPD

In the following figure, PQ = QR, ∠RQP  = 68°, PC and CQ are tangents to the circle with centre O.

Calculate the values of:

1. ∠QOP
2. ∠QCP

In the figure, given below, AC is a transverse common tangent to two circles with centres P and Q and of radii 6 cm and 3 cm respectively.

Given that AB = 8 cm, calculate PQ.

In the figure, given below, O is the centre of the circumcircle of triangle XYZ.

Tangents at X and Y intersect at point T. Given ∠XTY = 80° and ∠XOZ = 140°, calculate the value of ∠ZXY.

In the given circle with centre O, ∠ABC = 100°, ∠ACD = 40° and CT is a tangent to the circle at C. Find ∠ADC and ∠DCT.

In the figure given below, O is the center of the circle and SP is a tangent. If ∠SRT = 65°, find the value of x, y and Z.

(i) A tangent to a circle is a line coplanar with the circle which meets the circle at exactly one point.

(ii) When two circles touch each other at exactly one point, either externally or internally, they are said to be tangent to each other.

(a)

(b)

AB is the common tangent to both the circles that are tangent to each other.

Based on the above information, how many common tangents do you think can be drawn for two circles, when the circles are as given below? Draw the tangent(s) in each case.

(a)

(b)

(c)

(d)