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SUPERMIND STUDY, PUNE
8th MATHEMATICS
THE CIRCLE
IMPORTANT POINTS:
(1) The perpendicular drawn from the centre of the circle to a chord of the circle bisects
the chord.
(2) Congruent chords in the same circle or in the congruent circles are at an equal
distance from the centre of the circle.
(3) Congruent chords of the same circle or of the congruent circles form congruent
angles at the centre of the circle.
(4) The angle formed at the centre by a chord is called the central angle.
THE ARC OF A CIRCLE
IMPORTANT POINTS:
(1) Study the following figure:
(i) Intercepted arc: Arc AXC is in the interior of the L ABC and the endpoints of the
arc lie on the sides of L ABC. So, arc AXC is an intercepted arc and it is said to be
intercepted by angle L ABC. Similarly, arc AXC is also intercepted by L AOC.
(ii) Minor arc and major arc: Consider radii OA and OB divides, the circle into two
parts. Each part is called an arc.AXC is a
minor arc and arc AYC is a major arc.
(iii) The central angle: L AOC has centre O
as its vertex. So,
LAOC is a the central
angle.
(iv)The measure of a minor arc: The
central angle L AOC has intercepted arc
AXC. So L AOC corresponds to the minor
arc AXC. Therefore,
m (arc AXC) = m L AOC.
(v) The measure of a major arc: Arc AYC
is a major arc. Arc AXC is its corresponding
minor arc.
m (arc AYC) = 3600 – m (arc AXC)
(vi)The inscribed angle: Consider L ABC. Its vertex B is on the arc AYC. The
endpoints of arc AYC are A and C and they lie on the arms of L ABC. So, it is said
that L ABC is inscribed in the arc AYC.
(2) It must be remembered that a diameter divides the circle into to equal parts and
each part is called as a semicircle. The measure of a semicircle is 1800. The measure of
a circle is 3600.
(3) The measure of an inscribed angle is half the measure of the arc intercepted by it. In
1
1
m (arc AXC). In other words, m L ABC =
m L AOC.
the above figure: m L ABC =
2
2
(4) Cyclic quadrilateral: A quadrilateral whose all the four vertices are on the circle is
called a cyclic quadrilateral.
(5) The opposite angles of a cyclic quadrilateral are supplementary.
SUPERMIND STUDY, PUNE
9th GEOMETRY
CIRCLE
IMPORTANT THEOREMS:
(1) There is one and only one circle passing through the three non-collinear points.
(2) The perpendicular drawn from the centre of a circle to a chord bisects the chord.
(3) The segment joining midpoint of a chord and the centre of a circle is perpendicular to
the chord.
(4) Congruent chords of congruent circles are equidistant from the centre.
(5) Chords which are equidistant from the centre of congruent circles are congruent.
(6) Congruent chords of a circle subtend congruent angles at the centre.
(7) If the angles subtended by the chords at the centre of a circle are congruent then the
corresponding chords are congruent.
IMPORTANT POINTS:
(1) Concentric circles: The circles having same centre and different radii are called
concentric circles.
(2) Tangent: A line in the plane of the circle which intersects the circle in only one point
is called as the tangent of the circle.
(3) Secant: A line in the plane of the circle which intersects the circle in two distinct
points is called a secant.
(7) If the angles subtended by the chords at the centre of a circle are congruent then the
corresponding chords are congruent.
SUPERMIND STUDY, PUNE
10th GEOMETRY
CIRCLE
(A) PREPREPARATION:
(I) Important chapters for reference from 8th standard:
(1) The circle (2) The circumference and area of circle (3) The arc of a circle
(II) Important chapters for reference from 9th standard:
(2) Circle
(B) IMPORTANT DEFINITIONS, PROPERTIES AND FORMULAE:
(1) Tangent: A line in the plane of a circle which intersects the circle in one and only
one point is called a tangent of the circle and the point of intersection is called the point
of contact.
(2) Touching or tangent circles: Given two circles in a plane are said to be touching
circles or tangent circles if they have one and only one point in common.
(3) Inscribed angle: An angle is called inscribed angle in an arc of a circle if the vertex
of the angle is on the arc other than its end points and end points of the arc are on the
sides of the angle.
(4) Intercepted arc: Given an arc of the circle and an angle, if each side of the angle
contains an end point of the arc and all points of the arc except the end points are in the
interior of the angle, then the arc is said to be intercepted by the angle.
(5) Cyclic quadrilateral: A quadrilateral with all its vertices are on a circle is called
cyclic quadrilateral.
C) IMPORTANT THEOREMS AND COROLLARIES:
(1) A tangent at any point of a circle is perpendicular to the radius through the point of
contact.
(2) The line perpendicular to a radius of a circle at its outer end is tangent to the circle.
(3) The lengths of the two tangent segments to a circle drawn from an external point are
equal.
(4) If two circles are touching circles then the common point lies on the line joining their
centres.
(5) Inscribed angle theorem: This theorem can also be stated as: The measure of an
angle subtended by an arc at a point on the circle is half of the measure of the arc.
(6) An angle subtended by a semicircle at a point on the circle is a right angle.
(7) Angles subtended by the same arcs are congruent.
(8) The opposite angles of a cyclic quadrilateral are supplementary.
(9) If a pair of opposite angles of a quadrilateral is supplementary, then the quadrilateral
is cyclic.
(10) An exterior angle of cyclic quadrilateral is congruent to the angle opposite to its
adjacent interior angle.
(11) Tangent secant theorem: If an angle with its vertex
on the circle whose one side touches the circle and the
other intersects the circle in two points, then the measure
of the angle is half of the measure of its intercepted arc.
1
In the figure: L ABC = m (arc AEB)
2
(12) If a secant is drawn through the point of contact of a
tangent to a circle then the angles which this tangent
makes with the chord contained in the secant are equal
respectively to the angles subtended by the chord in
the corresponding alternate segments.
In the figure: (i) L AEB = L ABC and
(ii) L AFB = L ABD
(13) If a line is drawn intersecting to a secant of a circle
at a common point of the secant and the circle so that the angle formed by it with the
chord contained in the secant is equal to an angle subtended by the chord in the
alternate segment, then that line is a tangent to the circle.
(14) If two secants of a circle intersect
inside or outside the circle then the area of
the rectangle formed by the two line
segments corresponding to one
secant is equal in area to the rectangle
formed by the two line segments
corresponding to the other.
In the figure: AP x BP = CP x DP
(15) If a secant and a tangent of a circle intersect in a point outside the circle
then the area of the rectangle formed by the two line segments corresponding to the
secant is equal to the area of the square formed by the line segment corresponding to
the tangent.
In the figure: PA x PB = PT2
(D) SKILLS REQUIRED FOR THE MASTERY:
(1) Knowing all the theorems
(2) Keen observation and interpretation of geometrical
figures.
(E) SOLVED EXAMPLES:
(1) As shown in the figure, two circles intersect
each
other
in
two
points
A and B. Segment AB is the chord of both the
circles. Point C is the exterior point of both the
circles on the line AB. From the point C tangents are drawn to the circles
touching at M and N.
Prove that CM = CN.
Solution:
Proof: Line AC is a secant and line CM is a tangent.
∴ CM2 = CA x CB
[tangent secant property] ...... (1)
Line AC is a secant and line CN is a tangent.
∴ CN2 = CA x CB
[tangent secant property] ...... (2)
2
2
∴ CM = CN
[from (1) and (2)
∴ CM = CN
Hence proved.
(2) Two circles intersect each other in points A and B. Secants through A and B
intersect circles in C, D and M, N respectively. Prove that CM ||
DN.
Proof: □ ABMC is cyclic.
L ABN is its exterior angle and L ACM is the opposite angle of
adjacent interior angle of L ABN.
∴ L ABN = L ACM
...... (1)
□ ADNB is cyclic.
Opposite angles of a cyclic quadrilateral are supplementary. ∴ L ABN + L ADN =
1800.
∴ L ACM + L ADN = 1800.
[From (1)]
0
ie. L DCM + L CDN = 180 .
[C – A – D and M – B – N]
But these are interior angles and are proved to be supplementary.
∴ CM || DN
[Interior angles test]