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The center of the circle lies in______ of the circle.
The longest chord of the circle is:
Equal _____ of the congruent circles subtend equal angles at the centers.
Let and are two triangles inside the circle.
and (radii of the circle)
(Given)
So, (SSS congruency)
By CPCT rule, .
Hence, this prove the statement.
If chords AB and CD of congruent circles subtend equal angles at their centres, then:
In triangles and ,
(given)
and (radii of the circle)
So, . (SAS congruency)
(By CPCT)
If there are two separate circles drawn apart from each other, then the maximum number of common points they have:
The angle subtended by the diameter of a semicircle is:
The semicircle is half of the circle, hence the diameter of the semicircle will be a straight line subtending 180 degrees.
If AB and CD are two chords of a circle intersecting at point E, as per the given figure. Then:
(Equal chords are always equidistant from the centre)
(Common)
(perpendiculars)
So, (by RHS similarity criterion)
Hence, (by CPCT rule)
If a line intersects two concentric circles with centre O at A, B, C and D, then:
From the above fig., .
Therefore, ….(1)
Also, since bisects .
Therefore, ….(2)
From equation 1 and equation 2.
In the below figure, the value of \(\angle ADC \) is:
So,
An angle subtended by an arc at the centre of the circle is twice the angle subtended by that arc at any point on the rest part of the circle.
So,
In the given figure, find angle OPR.
The angle subtended by major arc PR at the centre of the circle is twice the angle subtended by that arc at point, Q, on the circle.
So, , here is the exterior angle
We know the values of angle as
So,
Now, in ,
OP and OR are the radii of the circle
So, OP = OR
Also,
By angle sum property of triangle, we know:
As,
Thus,
ABCD is a cyclic quadrilateral in which \(\angle DBC=55^o \) and \(\angle BAC=55^o \). Find \(\angle BCA \).
The radius of a circle is 2.5 cm. AB and CD are two parallel chords 2.7 cm apart. If AB = 4.8 cm. Find CD.
If \( \angle B=125^o\) and \( E\) is a point on the circle. Find \(\angle AEC \)
\(\triangle ABC\) is an isosceles triangle with \(AB = AC\) and \(\angle ABC=50^o \). Find the sum of \(\angle BDC \) and \(\angle BEC \)
(angles in the same segment are equal)
(Opposite angles of a cyclic quadrilateral are supplementary)
O is the centre of the circle. If \(\angle POQ=110^o \) and \(\angle POR=120^o \). Find \( \angle QPR\)
Find the value of y
The length of side QR is
Length of tangents from a outside point to the circle are equal.
\(\triangle ABC \) is inscribed in a circle. The bisector of \(\angle BAC \) meets BC at D and the circle at E. If EC is joined and \(\angle ECD=30^o \). Find \(\angle BAC \).
(angles in the same segment are equal)
(AD is a bisector)
48 cm long chord of a circle is at a distance of 7 cm from the centre. Find the radius of the circle.
Two circles with radii 6 and 12 are drawn with the same centre. The smaller inner circle is painted Pink and the part outside the smaller circle and inside the larger circle is painted green. What is the ratio of the areas painted green to the area painted pink?
If \(\angle ADB=80^o \) and \(\angle APB=125^o \). Find angle \(\angle DAC \).
(Linear pair)
O is the centre of a circle. If \(\angle DAC=54^o \) and \(\angle ACB=63^o \). Find \( \angle DAB\).
In
(Angle in a semicircle is )
If \(\angle BAD=100^o \) and \(\angle ACB=35^o \) Find angle \(\angle ABD \)
(angles inscribed by the same chord AB)
In
ABCD is a parallelogram. The circle through A, B, C intersect CD at E. Find the value of \(\dfrac{AE}{AD} \)
BC = AD ( ABCD is a parallelogram)
AE = BC (through construction)
So, AE = AD
A triangle ABC is inscribed in a circle, the bisector of whose angles meet the circumference at x, y, z. Determine the angles x, y, z
( angles in the same segment)
AB is a chord of a circle with centre o and radius 17 cm. If OM \(\bot \) AB and OM = 8 cm. Find the length of chord AB.
The length of tangent from point p to a circle of radius 5 cm is 12 cm. The distance p from the centre is
If \(\angle ADC=125^o \) and \(BC=BE \). Find \(\angle CBE \)
Two circles of radius 13 cm and 12 cm intersect at two points and the distance between their centers is 5 cm. Find the length of the common chord.
(radius)
(radius)
(common)
(SSS)
(cpct)
(cpct)
(cpct)
(Linear pair)
AC is a diameter of the circle and arc AXB = \( \dfrac{1}{3}\)ar BYC. Find \(\angle BOC \)
The ratio of arc length to the circumference
arc
Two circles with centre 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. Find the ratio of OO’ : PQ
PA is a chord and OR is a line perpendicular to it.
A perpendicular drawn from the centre of a circle bisects the chord.
PR = RA
PA = 2RA
Similarly AQ = 2 AS
A chord of a circle is 12 cm in length and its distance from the centre is 8 cm. Find the length of the chord of the same circle which is at a distance of 6 cm form the centre.
In
Radius
In
Length of the chord
Two chords AB and AC of a circle subtend angles equal to \(80^o \) and \(70^o \) respectively at the centre. Find \(\angle BAC \), if AB and AC lie on the opposite sides of the centre.
If BC is a diameter and \(\angle BAO=45^o \) Then the value of \(\angle ADC \) is
(radius)
is an isosceles triangle
(alternate angle)
The chords AB and CD of a circle are perpendicular to each other. If radius of the circle is 14 cm and length of the arc AQD is 24 cm. Find the length of arc BPC
arc RC = arc RD
arc RS = arc RC+ arc CS
arc RD + arc CS
arc RC+ arc SD
circumference = Length of arc BPC + length of arc AQD cm
Length of arc BPC cm