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In the figure, find the length of PR.
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In the figure, if BC = 4.5 cm, find the length of AB.
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A pair of tangents \(PA \) and \( PB\) are drawn from an external point \(P \) to a circle with centre \(O \). If \(\angle APB=90^o \) and \(PA=6\ cm \), find the radius of tehcircle.
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In
In the figure, BOA is a diameter of a circle and the tangent at point P meets BA produced at T. If \(\angle PBO=30^o \), find \(\angle \)PTA
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(radius)
In ,
The length of tangent from an external point on a circle is always greater than the radius of the circle. Is it true?
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False
It may be or may not be greater than the radius.
If the angle between two tangents drawn from a point P to a circle of radius a and centre O is 90°, then find OP.
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In the figure, circles C(O, r) and C’ (O’, r/2) touch ubternally at a point A and AB is a chord of the circle C (O, r), intersecting C’ (O’,r) at C. Prove that AC = CB.
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(angle in semicircle)
In and
and
(RHS)
(cpct) (Proved)
P is the midpoint of an arc QPR of a circle. Show that the tangent at P is parallel to the chord QR.
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Also by construction
So must be parallel to
In the figure, PQL and PRM are tangents to the circle with centre O at the points Q and R respectively and S is a point on the circle such that \(\angle SOL=50^o \) and \(\angle SRM=60^o \). Find \( \angle QSR\).
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In the figure, if \(\angle CBA=140^o \), find \(\angle ADB \).
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Show that the tangent to the circumcircle of an isosceles triangle ABC at A, in which AB = AC, is parrallel to BC.
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(given)
…(1)
is the tangent and is the chord
…(2)
From (1) and (2),
But they form alternate angles so
\(PQ \) and \(PR \) are tangent segment to a circle with centre O. If \(\angle QPR=80^o \), find \(\angle QOR \).
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\(AB \) is a diameter of a circle and \(AC \) is its chord such that \(\angle BAC=30^o \). If the tangent at \(C \) intersects \(AB \) at \(D \), then prove that \( BC=BD\).
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In
…(1)
In
Now
…(2)
…(3)
From (2) and (3)
In the figure, \(\angle ADC=90^o\ BC=38\ cm,CD=28\ cm \) and \(BP=25\ cm \). Find the radius of the circle.
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Let
is a square
If a number of circles pass through the end points P and Q of a line segment PQ, then prove that their centres lie on the perpendicular bisector of PQ.
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PQ is either a chord to a circle or the diameter.
If PQ is diameter than clearly centre lies on the left of diameter i.e. on the perpendicular bisector of PQ.
But if PQ is a chord then we know than perpendicular from the centre to the chord of circle bisect the chord implies that centre lies on the perpendicular bisector of the chord
So for each circle centre lies on the perpendicular bisector of PQ.
Two tangent segments \( BC\) and \(BD \) are drawn to a circle with centre \(O \) such that \(\angle CBD=120^o \). Show that \(OB=2BC \).
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In and
(Common)
(radius)
(RHS)
(cpct)
In the figure, \(\angle ADC=90^o\ BC=38\ cm,\ CD=28\ cm \) and \(BP=25\ cm \). Find the radius of the circle.
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Let
is a square
Out of the two concentric circles, the radius of the outer 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.
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In the isosceles triangle ABC shown below, AB = AC, show that BF = FC
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Since tangents from an exterior point to a cirlcle are equal
(given)
(Proved)
In the figure, a circle is inscribed in a \(\triangle \)ABC with sides AB = 12 cm, BC = 8 cm and AC = 10 cm. Find the lengths of AD, BE and CF.
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Let
Prove that opposite sides of quadrilateral circumscribing a circle subtend supplementary angles at the centre of circle.
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In
In
This opposite sides of a quadrilateral circumscribing a circle subtend supplementry angles at the centre of the circle
If all sides of a parallelogram touch a circle, then prove that it is a rhombus.
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…(1)
…(2)
…(3)
…(4)
Adding 1, 2, 3, 4 we get
( opposite sides are equal
But and
So
So it is a rhombus
Two tangents PA and PB are drawn from an external point P to a circle with centre. O. Prove that AOBP is a cyclic quadrialteral.
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and
..(1)
In quadrilateral
…(2)
From (1) and (2), the quadrilateral is cyclic.
Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord.
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In
(radius)
Now
This tangents make equal angle with the chord
In the figure, is the centre of two concentric circles of radii 6 cm and 4 cm. PQ and PR are tangents to the two circles from an external point P. If PQ = 10 cm, find the length of PR (upto one decimal place).
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In ,
Prove that the tangents drawn at the endpoints of a diameter of a circle are parallel.
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But (alt interior angles)
(alt interior angles)
Since alt are equal line and will be parallel.
In the figure, a quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.
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But
(Proved)
Prove that the line segment joining the points of contact of two parallel tangents to a circle is a diameter of the circle.
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Since (by construction)
Similarly
is a straight line so it is a diameter.
In the figure, \(O \) is the centre of a circle and \(BCD \) is tangent to it at \(C \). Prove that \(\angle BAC+\angle ACD=90^o \).
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(radius)
…(1)
In the figure, two circles touch each other externally at C. Prove that the common tangent at C bisects the other two common tangents.
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We know that tangent are equal in length from an external point to circle.
and and
Hence bisects and bisects
In the figure, a circle touches the side BC of triangle ABC at P and touches AB and AC produced at Q and R respectively. Show that AQ=\(\dfrac{1}{2} \) (Perimeter of \(\triangle \)ABC )
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In the figure, all three sides of a triangle touch the circle. Find the value of x.
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Two tangents PA and PB are drawn to a circle with centre O, such that \(\angle APB=120^o \). Prove that OP = 2 AP.
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So
In
In the figure, O is the centre of the circle. PA and PB are tangents to the circle from the point P prove that AOBP is a cyclic quadrilateral.
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In quadrilateral
Since the sum of the opposite angles of quadrilateral is
Hence is a cyclic quadrilateral
In the figure, CP and CQ are tangents to a circle with centre O. ARB is another tangent touching the circle at R. If CP = 11 cm, and BC = 7 cm, then find the length of BR.
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In the figure, \(\triangle \)ABC is circumscribing a circle. Find the length of BC.
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So
A tangent PT is drawn parallel to a chord AB as shown in figure. Prove that APB is an isosceles triangle.
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( is a bisector of )
In and
(each 90)
(common)
(SAS)
(cpct)
is an isosceles
Two tangents PA and PB are drawn from an external point P to a circle with centre O. Prove that AOBP is a cyclic quadrilateral.
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Since, and
is a cyclic quadrilateral.
In the figure, AB and CD are common tangents to two circles of unequal radii. Prove that AB = CD.
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If \(d_1,d_2\ (d_2>d_1) \) be the diameters of two concentric circles and c be the length of a chord of a circle, which is tangent to the other circle, prove that \(d^2_2=c_2+d^2_1 \).
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(as C bisects AB)
ABC is a right triangle, right angled at A. A circle is inscribed in it. The lengths of the sides containing the right angle are 8 cm and 6 cm. Find the radius of the incircle.
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Area of = Area of + Area of + Area of
a, b, c are the sides of a right triangle, where c is the hypotenune. Prove that the radius r of the circle which touches the sides of the triangle is given by \(r=\dfrac{a+bc}{2} \)
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AB is a diameter of a circle APB. AH and BK are perpendicular from A and B respectively to the tangent at P. Prove that AB + AH = BK.
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Let
(AA)
…(1)
Also (AA)
..(2)
From (1) and (2)
(Proved)
Show that the tangents drawn at the ends of chord of a circle make equals angles with the chord.
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In
(radius)
Now,
Tangents make equal angles with the chord.
Let A be the one point of intersection of two intersecting circles with centre O and Q. The tangents at A to the two circles meet the circles again at B and C respectively. Let the point P be located such that AOPQ is a parallelogram. Prove that P is the circumcentre of \(\triangle \)ABC .
In \(\triangle ABC \)
\( PA=PB\)
Similarly \(PA=PC \)
Hence \(PA=PB=PC \)
Hence \(P \) is equdratant from the 3 vertices of \(\triangle ABC \).
So \(P \) is the circumcentre of \( \triangle ABC\)
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Perpendicular from the centre of circle to a chord bisects the chord
This and is the perpendicular bisector of . Similarly is the perpendicular bisector of .