###### Advertisements

###### Advertisements

Out of the two concentric circles, the radius of the outer circle is 5 cm and the chord AC of length 8 cm is a tangent to the inner circle. Find the radius of the inner circle.

###### Advertisements

#### Solution

From the figure,

Chord AB = 8 cm

OC is perpendicular to the chord AB

AC = CB = 4 cm

In right triangle OCA

OC^{2} + CA^{2} = OA^{2}

OC^{2} = 5^{2 }– 4^{2}

= 25 – 16

= 9

OC = 3 cm

#### APPEARS IN

#### RELATED QUESTIONS

A tangent to a circle intersects it in ______ point (s).

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

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

In Fig. 1, AOB is a diameter of a circle with centre O and AC is a tangent to the circle at A. If ∠BOC = 130°, the find ∠ACO.

In Figure 2, XP and XQ are two tangents to the circle with centre O, drawn from an external point X. ARB is another tangent, touching the circle at R. Prove that XA + AR = XB + BR ?

In the adjoining figure the radius of a circle with centre C is 6 cm, line AB is a tangent at A. Answer the following questions.

(1) What is the measure of ∠CAB ? Why ?

(2) What is the distance of point C from line AB? Why ?

(3) d(A,B) = 6 cm, find d(B,C).

(4) What is the measure of ∠ABC ? Why ?

What is the distance between two parallel tangents of a circle having radius 4.5 cm ? Justify your answer.

Four alternative answers for the following question is given. Choose the correct alternative.

If two circles are touching externally, how many common tangents of them can be drawn?

How many common tangents can be drawn to two circles, touching each

other externally?

Draw Δ ABC such that, AB = 8 cm, BC = 6 cm and ∠ B = 90°. Draw seg BD

perpendicular to hypotenuse AC. Draw a circle passing through points

B, D, A. Show that line CB is a tangent of the circle.

In fig., AB and DC are two chords of a circle with centre O. these chords when produced meet at P. if PB = Bern, BA = 7cm and PO = 14.5cm, find the radius of the circle.

In following fig., PT is a tangent to the circle at T and PAB is a secant to the same circle. If PB = 9cm and AB = Scm, find PT.

PA and PB are tangents from P to the circle with centre O. At M, a tangent is drawn cutting PA at K and PB at N. Prove that KN = AK + BN.

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 AB.

In a square ABCD, its diagonal AC and BD intersect each other at point O. The bisector of angle DAO meets BD at point M and bisector of angle ABD meets AC at N and AM at L. Show that - ∠ BAM = ∠ BMA

Construct a tangent to a circle with centre O and radius 3.5 cm at a point P on it.

In the following figure, seg AB is a diameter of the circle, m (arc AKC) = 40°. Find the value of m (arc BMC).

In the following figure, seg AB is the diameter of the circle with center P. Line CB be the tangent and line AC intersects a circle in point D. Prove that:

AC x AD = 4 (radius)^{2}

^{}

In Fig. TA is a tangent to a circle from the point T and TBC is a secant to the circle. If AD is the bisector of ∠BAC, prove that ΔADT is isosceles.

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

find: (i) ∠ BCO (ii) ∠ AOB (iii) ∠ APB

Draw a circle of radius 2.7 cm and draw a chord PQ of length 4.5 cm. Draw tangents at points P and Q without using centre.

In figure, M is the centre of the circle and seg KL is a tangent segment. If MK = 12, KL = `6sqrt(3)`, then find

(i) Radius of the circle.

(ii) Measures of ∠K and ∠M.

In figure, if O is the centre of a circle PQ is a chord and the tangent PR at P makes an angle of 50° with PQ, then ∠POQ is equal to ______.

In figure, if PQR is the tangent to a circle at Q whose centre is O, AB is a chord parallel to PR and ∠BQR = 70°, then ∠AQB is equal to ______.

Tangents from an external point to a circle are ______

In the figure, PQ and PR are tangents to a circle with centre A. If ∠QPA=27°, then ∠QAR equals to ______

In the given figure, PT is a tangent to the circle at T, chord BA is produced to meet the tangent at P. Perpendicular BC bisects the chord TA at C. If PA = 9 cm and TB = 7 cm, find the lengths of:

- AB
- PT

In the above figure, seg AB and seg AD are tangent segments drawn to a circle with centre C from exterior point A, then prove that: ∠A = `1/2` [m(arc BYD) - m(arc BXD)]

A tangent JK is drawn to a circle with centre C such that CK = 6 cm and ∠CKJ = 60°. Find the length of the tangent JK.

The distance between two tangents parallel to each other of a circle is 13 cm. Find the radius of the circle.

In the adjoining figure, PT is a tangent at T to the circle with centre O. If ∠TPO = 30°, find the value of x.

In the given figure, QR is a common tangent to the two given circles touching externally at A. The tangent at A meets QR at P. If AP = 4.2 cm, then the length of QR is ______.