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A circle with centre O and radius 5 cm has been inscribed in an equilateral triangle ABC. Find the perimeter of \(\triangle \)ABC .
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it is an equilateral triangle
Area of equilateral
Area of = Area of + Area of + Area of
Perimeter
A chord PQ of a circle is parallel to the tangent drawn at a point R of the circle. Prove that R bisects the arc PRQ.
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In and
(radius)
(common)
(RHS)
(cpct)
This bisects
In the figure, common tangents AB and CD to two circles intersect at E. Prove that AB = CD.
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(E is the external point)
(E is the external point)
Adding eqn (1) are (2)
(Proved)
Prove that the centre of a circle touching two intersecting lines lies on the angles bisector of the lines.
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(radius)
(radius is perpendicular to its tangent)
(common)
(SAS)
(cpct)
bisects
lies on the bisector of the angle between and
Prove that there is one and only one tangent at any point on the circumference of a circle.
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Let point P on the circle
If posible let PT’ be the tangent instead of PT
and
This is only possible if PT’ concide PT
Therefore there is only one tangent to a point on circle.
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
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(radii)
is an arbitrary point on the tangent
Thus is shorter than any other line segment joining O to any point on tangent. Shortest distance of a point from a given line is the perpendicular distance from that line
Hence the tangent at any point of circle is perpendicular to the radius
In the figure, I and m are two parallel tangents at A and B. The tangent at C makes an intercept DE between l and m. Prove that DE subtends a right angle at the centre of the circle.
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In and
(radii)
(length of tangent )
(common)
(SSS)
Similarly
is a diameter of the circle
Hence it is a straight line
In the figure, triangle \(ABC \) is a right angled triangle with \(AB = 6\ cm, AC = 8\ cm\ \angle A=90^o\) . A circle with centre \(O \) is inscribed inside the triangle. Find the radius r.
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Area of = area of
If the an external point B of the circle with centre O, two tangents BC and BD are drawn such that \(\angle DBC=120^o \), prove that BO = 2BC.
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In and
(common)
(radius)
(RHS)
(cpct)
In the figure, from an external point P, PA and PB are tangent to the circle with centre O. If CD is another tangent at point E to the circle and PA = 12 cm, find the perimeter of \(\triangle \)PCD .
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Perimeter of
In the figure, OP is equal to the diameter of the circle. Prove that ABP is an equilateral triangle
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In
Similarly
Since (length of tangent from an external point are equal)
In
Since all angles are is equilateral
In the figure, a circle touches all the four sides of a quadrilateral ABCD with sides AB = 6 cm, BC = 7 cm and CD = 4 cm. Find AD.
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Adding all the equation we get
A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact are of lengths 8 cm and 6 cm respectively.
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Area of
â€¦(1)
Area of = area of …(2)
From eqn (1) and (2)
A circle touches the side BC of \(\triangle \)ABC at P and sides AB and AC produced at Q and R respectively. Prove that \(AQ=\dfrac{1}{2} \) (Perimeter of \(\triangle \)ABC)
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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= (Perimeter of ABC )
In the figure, triangle ABC is isosceles in which AB = AC, circumscribed about a circle. Prove that base is bisected by the point of contact.
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is an isosceles )
So is bisected at the point of contact
Prove that the angle between the two tangents to a circle drawn from an external point, is supplementary to the angle subtended by the line segment joining the points of contact at the centre
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If proves that angle between the two tangents drawn from an external point to a circle supplementry to the angle subtended by the line segment.
Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that \(\angle APB=2\ \angle OAB \).
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..(1)
is an isosceles triangle since
….(2)
In the figure, XP and XQ are tangents from an external point X to the circle with centre O. R is a point on the circle where another tangent ARB is drawn to the circle. Prove that XA + AR = XB + BR.
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(from X) â€¦(1)
(from A) â€¦.(2)
(from B) â€¦.(3)
(using eqn (2) and (3))
In the figure, PO \(\bot \) QO. The tangents to the circle with centre O at P and Q intersect at a point T. Prove that PQ and OT are right bisectors of each other.
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(radius)
â€¦(1)
â€¦(2)
â€¦(3)
From (1), (2), (3)
form a square and bisector of square bisector each other at 90.
In the figure, PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at point P and Q intersect at point T. Find the length of tangent TP.
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Let
Since is the perpendicular bisector of
Also
In the figure, ABC is a right triangle, right angled at A, with AB = 6 cm and AC = 8 cm. A circle with centre O has been inscribed inside the triangle. Calculate the radius of the inscribed circle.
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Area of = area of
In the figure, \(PT \) and \(PS \) are tangents to a circle from a point \(P \) such that \(PT=5\ cm \) and \(\angle TPS=60^o \). Find the length of chord \(TS \).
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\(PAQ \) is a tangent to the circle with centre \(O \) at a point \(A \) as shown in the figure. If \(\angle OBA=35^o \), find the value of \(\angle BAQ \) and \(\angle ACB \).
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But
(Angles in the same segment)
In the figure, a circle is inscribed in a quadrilateral \(ABCD \) in which \(\angle B=90^o \). If \(AD=23\ cm,\ AB=29\ cm \) and \(DS=5\ cm \), find the radius (r) of the circle
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So
Also
In quadrilatral
(given)
(angle between the radius and tangent at point of contact)
is a square
So radius = 11 cm
In the figure, OP is equal to diameter of the circle. Prove that ABP is an equilateral triangle.
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Let
In
Similarly
Since
In ,
Since all angle are , So is an equilateral
Two circles with centres O and Oâ€™ of radii 3 cm and 4 cm respectively intersect at two points P and Q such that OP and Oâ€™ P are tangents to the two circles. Find the length of the common chord PQ.
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Let , then
Prove that the tangent drawn at the midpoint of an arc of a circle is parallel to the chord joining the end points of the arc.
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Since P is the mid point of arc APB
In and
(common)
(Proved above)
(SAS)
(cpct)
But they are corresponds angle
So
In a right triangle ABC, in which \(\angle B=90^o \), a circle is drawn with AB as diameter intersecting the hypotenuse AC at P. Prove that the tangent to the circle at P bisects BC.
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(angle between tangent and the chord equals angle made by the chord in alternate segment)
â€¦(1)
(angle in semicircle)
…(2)
From (1) and (2)
Also (tangent drawn from an internal point are equal )
(Proved)
From a point P, two tangents PA and PB are drawn to a circle C (O, r). If OP = 2r, show that \(\triangle \)APB is equilateral.
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In
Similarly it can be proced that
In ,
is an equilateral
O is the centre of a circle, PA and PB are tangents to the circle from a point P. Prove that (i) PAOB is a cyclic quadrilateral (ii) PO is the bisector of \(\angle \)APB. (iii) \(\angle \)OAB = \(\angle \)OPA
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(i)
Since the sum of the opposite angles of the quadrilateral is
Hence is a cyclic quadrilateral
(ii) In and
(each 90)
(common)
(radius)
(RHS)
(cpct)
is the bisector of
If a circle touches the side BC of a triangle ABC at P and extended sides AB and AC at Q and R respectively, prove that \(AQ=\dfrac{1}{2}(AB+BC+CA) \)
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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= (Perimeter of ABC )
If an isosceles triangle ABC, in which AB = AC = 6 cm is inscribed in a circle of radius 9 cm, find the area of the triangle.
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Let
is an isosceles
Area of
In the figure, from an external point P, a tangent PT and a line segment PAB is drawn to a circle with centre O. ON is perpendicular to the chord AB. Prove that :
(a) \(PA. PB = PN^2 – AN^2 \)
(b) \(OP^2 – OT^2 = PN^2 – AN^2 \)
(c) \( PA. PB = PT^2.\)
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(i)
(ii)
From parts (1) and (2)
If a hexagon ABCDEF circumscribes a circle, prove that AB + CD + EF = BC + DE + FA.
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â€¦.(1)
â€¦(2)
â€¦(3)
â€¦(4)
â€¦(5)
â€¦(6)
Adding (1) and (2)
Adding (3) and (4)
Adding (5) and (6)
Adding we obtain
In the figure, O is the centre of the circle. If OR = 5 cm and OA = 13 cm, find the perimeter of \(\triangle \)ABC
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Perimeter of
Perimeter of
The transverse common tangents AB and CD of two circles with centre O and Oâ€™ intersect at E. Prove that the points O, E and Oâ€™ are collinear.
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In and
Similarly
Since there angles are equal and are bisected by OE and O’E.
O, E, O’ are collinear
Let s denote the semi perimeter of a \(\triangle \)ABC in which BC = a, CA = b and AB = c. If a circle touches BC, CA and AB at D, E and F respectively, prove that BD = s â€“ b.
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Let
Perimeter of
In the figure, tangents \(PQ \) and \(PR \) are drawn to a circle such that \(\angle RPQ=30^o \). A chord \(RS \) is drawn parallel to the tangents \(PQ \). Find \( \angle RQS\).
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In
(Alternate segment theorem)
(alt )
In the figure, the tangent at a point \(C \) of a circle and a diameter \(AB \) when extended intersect at \( P\). If \(\angle PCA=110^o \), find \(\angle CBA \).
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(Angle in a semicircle)
In
OABC is a rhombus whose three vertices A,B and C lie on a circle with centre O. If the radius of the circle is 10 cm, find the area of the rhombus.
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is a rhombus
is an isosceles
Area of rhombus
Prove that the lengths of the tangents drawn from an external point to a circle are equal.
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In and
(radius)
(common)
(RHS)
(cpct)
Thus it is proved that the length of the two tangents drawn from an external point to a circle are equal.
Prove that the angle between the two tangents drawn from any external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.
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AS radius of the circle is perpendicular to the tangent
Similarly
It proves that the angle between the two tangents drawn from an external point to a circle supplementry to the angle subtended by the line segment.
A circle touches the sides of a quadrilateral ABCD at P, Q R, S respectively. Show that angle subtended at the centre by pairs of opposite sides are supplementary.Using the above, find \(\angle \)PTQ in the figure. If TP and TQ are the two tangents to a circle with centre O so that \(\angle POQ=110^o \).
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Since the two tangents drawn from an external point to a circle subtend equal angles at the centre
and and
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. Using the above , do the following :In figure, O is the centre of the two concentric circles. AB is a chord of the larger circle touching the smaller circle at C. Prove that AC = BC.
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(radii)
(radius is )
So (as perpendicular from the centre bisects the chord)