A chord of length 16cm is drawn in a circle of radius 10cm. Find the distance of the chord from the centre of the circle.
Find the length of a chord which is at a distance of 3cm from the centre of a circle of radius 5cm.
A chord of length 30cm is drawn at a distance of 8cm from the centre of a circle. Find out the radius of the circle.
In a circle of radius 5cm, AB and CD are two parallel chords of lengths 8cm and 6cm respectively. Calculate the distance between the chords if they are
(i) on the same side of the centre,
(ii) on the opposite sides of the centre.
Two parallel chords of lengths 30cm and 16cm are drawn on the opposite sides of the centre of a circle of radius 17cm. Find the distance between the chords.
In the given figure, the diameter CD of a circle with centre O is perpendicular to chord AB. If AB = 12cm and CE = 3 cm, calculate the radius of the circle.
In the given figure, a circle with centre O is given in which a diameter AB bisects the chord CD at a point E such that CE = ED = 8cm and EB = 4cm. Find the radius of the circle.
In the adjoining figure, OD is perpendicular to the chord AB of a circle with centre O. If BC is a diameter, show that AC || DO and AC = 2 × OD.
In the given figure, O is the centre of a circle in which chords AB and CD intersect at P such that PO bisects ∠BPD. Prove that AB = CD.
Prove that the diameter of a circle perpendicular to one of the two parallel chords of a circle is perpendicular to the other and bisects it.
Prove that two different circles cannot intersect each other at more than two points.
Two circles of radii 10 cm and 8 cm intersect each other, and the length of the common chord is 12 cm. Find the distance between their centres.
Two equal circles intersect in P and Q. A straight line through P meets the circles in A and B. Prove that QA = QB.
If a diameter of a circle bisects each of the two chords of a circle then prove that the chords are parallel.
In the adjoining figure, two circles with centres at A and B, and of radii 5cm and 3cm touch each other internally. If the perpendicular bisector of AB meets the bigger circle in P and Q, find the length of PQ.
In the given figure, AB is a chord of a circle with centre O and AB is produced to C such that BC = OB. Also, CO is joined and produced to meet the circle in D. If ∠ACD = y and ∠AOD = x, prove that x = 3y.
AB and AC are two chords of a circle of radius r such that AB = 2AC. If p and q are the distances of AB and AC from the centre then prove that 4q^2 = p^2 + 3r^2.
In the adjoining figure, O is the centre of a circle. If AB and AC are chords of the circle such that AB =
AC, OP ⊥ AB and OQ ⊥ AC, prove that PB = QC.
In the adjoining figure, BC is a diameter of a circle with centre O. If AB and CD are two chords such that AB || CD, prove that AB = CD.
An equilateral triangle of side 9cm is inscribed in a circle. Find the radius of the circle.
In the adjoining figure, AB and AC are two equal chords of a circle with centre O. Show that O lies on the bisector of ∠BAC.
In the adjoining figure, OPQR is a square. A circle drawn with centre O cuts the square in X and Y. Prove that QX = QY.
Two circles with centres O and O’ intersect at two points A and B. A line PQ is drawn parallel to OO’ through A or B, intersecting the circles at P and Q. Prove that PQ = 2OO’.
(i) In Figure (1), O is the centre of the circle. If ∠OAB = 40 and ∠OCB = 30, find ∠AOC.
(ii) In Figure (2), A, B and C are three points on the circle with centre O such that ∠AOB = 90 and ∠AOC = 110. Find ∠BAC.
In the given figure, O is the centre of the circle and ∠AOB = 70. Calculate the values of
(i) ∠OCA,
(ii) ∠OAC.
In the given figure, O is the centre of the circle. If ∠PBC = 25 and ∠APB = 110, find the value of ∠ADB.
In the given figure, O is the centre of the circle. If ∠ABD = 35 and ∠BAC = 70, find ∠ACB.
In the given figure, O is the centre of the circle. If ∠ACB = 50, find ∠OAB.
In the given figure, ∠ABD = 54 and ∠BCD = 43, calculate
(i) ∠ACD,
(ii) ∠BAD,
(iii) ∠BDA.
In the adjoining figure, DE is a chord parallel to diameter AC of the circle with centre O. If ∠CBD = 60,calculate ∠CDE.
In the adjoining figure, O is the centre of a circle. Chord CD is parallel to diameter AB. If ∠ABC = 25, calculate ∠CED.
In the given figure, AB and CD are straight lines through the centre O of a circle. If ∠AOC = 80 and ∠CDE = 40, find
(i) ∠DCE,
(ii) ∠ABC.
In the given figure, O is the centre of a circle, ∠AOB = 40 and ∠BDC = 100, find ∠OBC.
In the adjoining figure, chords AC and BD of a circle with centre O, intersect at right angles at E. If ∠OAB = 25, calculate ∠EBC.
In the given figure, O is the centre of a circle in which ∠OAB = 20 and ∠OCB = 55. Find
(i) ∠BOC,
(ii) ∠AOC.
In the given figure, O is the centre of the circle and ∠BCO = 30. Find x and y.
In the given figure, O is the centre of the circle, BD = OD and CD ⊥ AB. Find ∠CAB.
In the given figure, PQ is a diameter of a circle with centre O. If ∠PQR = 65, ∠SPR = 40 and ∠PQM = 50, find ∠QPR, ∠QPM and ∠PRS.
In the figure given below, P and Q are centres of two circles, intersecting at B and C, and ACD is a straight line. If ∠APB = 150 and ∠BQD = x, find the value of x.
In the given figure, ∠BAC = 30. Show that BC is equal to the radius of the circumcircle of △ ABC whose centre is O.
In the given figure, AB and CD are two chords of a circle, intersecting each other at a point E. Prove that ∠AEC = ½ (angle subtended by arc CXA. At the centre + angle subtended by arc DYB at the centre).
In the given figure, ABCD is a cyclic quadrilateral whose diagonals intersect at P such that ∠DBC = 60 and ∠BAC = 40. Find
(i) ∠BCD,
(ii) ∠CAD
In the given figure, POQ is a diameter and PQRS is a cyclic quadrilateral. If ∠PSR = 150, find ∠RPQ.
In the given figure, O is the centre of the circle and arc ABC subtends an angle of 130 at the centre. If AB is extended to P, find ∠PBC.
In the given figure, ABCD is a cyclic quadrilateral in which AE is drawn parallel to CD, and BA is produced to F. If ∠ABC = 92 and ∠FAE = 20, find ∠BCD.
In the given figure, BD = DC and ∠CBD = 30, find ∠BAC.
In the given figure, O is the centre of the given circle and measure of arc ABC is 100. Determine ∠ADC and ∠ABC.
In the given figure, △ ABC is equilateral. Find
(i) ∠BDC,
(ii) ∠BEC.
In the adjoining figure, ABCD is a cyclic quadrilateral in which ∠BCD = 100 and ∠ABD = 50. Find ∠ADB.
In the given figure, O is the centre of a circle and ∠BOD =150. Find the values of x and y.
In the given figure, O is the centre of the circle and ∠DAB = 50. Calculate the values of x and y.
In the given figure, sides AD and AB of cyclic quadrilateral ABCD are produced to E and F respectively. If ∠CBF = 130 and ∠CDE = x, find the value of x
In the given figure, AB is a diameter of a circle with centre O and DO || CB. If ∠BCD = 120, calculate
(i) ∠BAD,
(ii) ∠ABD,
(iii) ∠CBD,
(iv) ∠ADC.
Also, show that △ AOD is an equilateral triangle
Two chords AB and CD of a circle intersect each other at P outside the circle. If AB = 6cm, BP = 2cm and PD = 2.5cm, find CD.
In the given figure, O is the centre of a circle. If ∠AOD = 140 and ∠CAB = 50, calculate
(i) ∠EDB,
(ii) ∠EBD
In the given figure, △ ABC is an isosceles triangle in which AB = AC and a circle passing through B and C intersects AB and AC at D and E respectively. Prove that DE || BC.
In the given figure, AB and CD are two parallel chords of a circle. If BDE and ACE are straight lines, intersecting at E, prove that △ AEB is isosceles.
In the given figure, ∠BAD = 75o, ∠DCF = xo and ∠DEF = yo. Find the values of x and y.
In the given figure, ABCD is a quadrilateral in which AD = BC and ∠ADC = ∠BCD. Show that the points A, B, C, D lie on a circle.
Prove that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent.
Prove that the circles described with the four sides of a rhombus as diameters pass through the point of intersection of its diagonals.
ABCD is a rectangle. Prove that the centre of the circle through A, B, C, D is the point of intersection of its diagonals.
Give a geometrical construction for finding the fourth point lying on a circle passing through three given points, without finding the centre of the circle. Justify the construction.
In a cyclic quadrilateral ABCD, if (∠B – ∠D) = 60, show that the smaller of the two is 60.
The diagonals of a cyclic quadrilateral are at right angles. Prove that the perpendicular from the point of their intersection on any side when produced backwards, bisects the opposite side.
On a common hypotenuse AB, two right triangles ACB and ADB are situated on opposite sides. Prove that ∠BAC = ∠BDC.
ABCD is a quadrilateral such that A is the centre of the circle passing through B, C and D. Prove that ∠CBD + ∠CDB = ½ ∠BAD.
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