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Can we have a cyclic quadrialateral \(ABCD \) such that \(\angle A=90^o,\angle B=70^o,\angle C=95^o \) and \(\angle D=105^o \)?
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False
Sum of opposite angles is not equal to . So it is not a cyclic quadrialateral
Two congruent circles with centres \(O \) and \(O’ \) intersect at two points \( A\) and \(B \). Check whether \(\angle AOB=\angle AO’B \) or not.
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In and
(Radius)
(Radius)
(common)
(cpct)
In the figure, PQR is right angled at Q. Point S is taken on side PR such that PS = SR and QR = QS. Find the measure of \(\angle \)QSR
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(Radius)
But (given) and (given)
So is an equlateral
So
Show that diameter of a circle is the greatest chord.
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Take any chord in a circle with end points . Let be the centre of the circle. Then segments and are ratio of the circle is a triangle.
(Sum of two sides is greater then the third side)
The maximum length of chord is
In the figure, \(\triangle \)ABC is equilateral. Find \(\angle \)BDC and \(\angle \)BEC
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(angle made by same arc are equal)
(Sum of opposite vertices of cyclic qaudrilateral is )
In the figure, if \(AOB \) is a diameter and \(\angle ADC=120^o \), find \(\angle CAB \)
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In
If arcs AXB and CYD of a circle are congruent, find the ratio of AB and CD
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Let and are arcs of cirlces whose centre and radius are and units. Hence ratio of and is
Show that two circles cannot intersect at more than two points.
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Since 2 distinct circles intersect at more than 2 points. These points are noncollinear points as 3 non collinear points determine one and only one circle, there should be only one circle. This contradicts assumption, so 2 circles cannot intersect each other at more than 2 points.
Show that the altitudes of a triangle are concurrent.
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(Since RACB a parallelogram)
(ABCD is a parallelogram)
…(1)
Since is
is the perpendicular bisector of
Similarly and is the perpendicular bisector of and respectively.
and are concurrent as we know that the perpendicular bisector of the sides of a triangle are concurrent at a point called circumcenter.
Prove that angle bisector opposite angle of a triangle and the perpendicular bisector of the opposite side if intersect, they will interect on the circumcircle of the triangle.
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(Angles in the same segment)
…(1)
( is bisector of )
Similarly …(2)
From (1) and (2)
lies on perpendicular bisector of
ABC is an equilateral triangle inscribed in a circle and P be any point on the minor are BC which does not coincide with B or C. Prove that PA is angle bisector of \(\angle \)BPC.
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( is equalatiral )
(equal chord subtend equal angles at centre)
…(1)
…(2)
…(3)
So, (Proved)
Prove that the angles in a segment greater than a semi–circle is less than a right angle.
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(central angle is twice of inscribed angle)
(Proved)
If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection. Prove that the chords are equal.
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In and
(each )
(given)
(common)
(AAS)
(cpct)
(chords equidistant from the centre are equal)
AB and AC are two chords of a circle of radius r units. If AB = 2AC, and the length of the perpendicular from the centre on these chords are a and b respectively, prove that \(4b^2=a^2+3r^2 \).
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In
…(1)
In
…(2)
From (1) and (2)
P is the centre of the circle. Prove that \(\angle XPZ=2\ [\angle XYZ+\angle XZY] \).
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…(1) (central angle is twice of inscribed angle)
…(2) (central angle is twice of inscribed angle)
Adding equation (1) and (2)
(Proved)
Prove that the angle subtended by an arc at the centre is double the angles subtended by it at any point on the remaining part of the circle.
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In
(Radius) …(1)
(exterior angle is equal to the sum of interior opposite angle)
…(2)
Similarly ….(3)
Adding (2) and (3)
…(4)
In case 3
Relax angle (Proved)
In the figure. \(ABCD \) is a cyclic quadrilateral, \(O \) is the centre of the circle. If \( \angle BOD=160^o\), find \(\angle BPD \).
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Since is a cyclic quadrilatral
(Since angles on the same segment are equal)
In the figure, \(ABCD \) is a cyclic quadrilateral in which \(AB \) is produced to \(F \) and \(BEDC \). If \(\angle FBE=20^o \) and \(\angle DAB=95^o \), find \( \angle ADC\).
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(opposite angles of a cyclic quadrilateral)
(alt )
In the figure, B and E are points on the line segment AC and DF respectively. Show that AD  CF.
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…(1) (opposite of a cyclic quadrilateral)
(Linear pair) …(2)
From (1) and (2)
(Since the cointerior angles are supplementry)
If O is the centre of a circle as shown in the given figure, then prove that x + y = z.
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In
…(1)
In
….(2)
( angles in the same segment)
Two equal chords AB and CD of a circle when produced intersect at the point P. Prove that PB = PD.
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(equal chords are equidistant from the centre)
In and
(each 90)
(common)
(RHS)
(cpct)…(1)
…(2)
In the figrue, find the values of a, b, c and d. Given \(\angle BCD=43^o\) and \(\angle BAE=62^o \).
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(exterior angle of a cyclic quadrilateral is equal to its interior opposite angle)
…(1)
In
…(2)
In
Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively. Prove that \(\angle \)ACP = \(\angle \)QCD.
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(Angles on the same segment) …(1)
(Angles on the same segment) …(2)
(VOA)
So,
Prove that the circle drawn on any one of the equal sides of an isosceles triangle as diameter bisects the base of the triangle.
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(angle in a semicircle is 90)
(Linear pair)
In and
(equal sides of isosceles triangle)
(each 90)
(common)
(RHS)
(Proved)