Hg
Name: Class 9
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The Hg Classes (8
th
to 12
th
) By: Er Hershit Goyal (B.Tech. IIT BHU), 134-SF, Woodstock Floors, Nirvana Country, Sector 50, GURUGRAM +91 9599697178.
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CIRCLES
1.
A circle is a collection of all points in a plane, which are equidistant from a fixed point.
common distance = radius; fixed point = centre.
2.
Equal chords of a circle subtend equal angles at the centre.
i.e. If AB = CD, then ∠AOB = ∠COD.
Converse: If 2 chords subtend equal angles at the centre, they are equal.
i.e. If ∠AOB = ∠COD, then AB = CD.
3.
Perpendicular from centre to the chord bisects the chord.
i.e. If OX ⟂ AB, then AX = XB.
Converse: Line joining the centre to the mid-point of a chord, is
perpendicular to the chord. i.e. If AX = XB, then OX ⟂ AB.
4.
Equal chords of a circle are equidistant from the centre.
i.e. If AB = CD, then OX = OY (where OX ⟂ AB and OY ⟂ CD).
Converse: Chords equidistant from the centre are equal.
i.e. If OX = OY (where OX ⟂ AB and OY ⟂ CD), then AB = CD.
# 2, 4 and their converse also apply on congruent circles.
5.
There can be infinite circles passing through 1 point or 2 points. But, there is a unique circle passing
through 3 non-collinear points.
6.
Angle subtended by a chord (or arc) at
centre is double the angle subtended
at circumference.
7.
Angles in the same
segment are equal.
8.
Angle in a semi-circle = 90
o
.
9.
If a line segment joining two points subtend equal angles at the other two points lying on the same
side of the line segment, then the four points are con-cyclic (i.e. lie on a circle).
10.
Sum of opposite angles of a cyclic quadrilateral is 180
o
.
i.e. ∠a + ∠c = 180, ∠b + ∠d = 180
Converse: If sum of opposite angles of a quad. is 180
o
, then it is cyclic.
Exterior angle of a cyclic quad. = interior opposite angle. i.e. ∠e = ∠a.
