Circle Notes for SSC CGL and CHSL - SSCADDA

Circle Notes for SSC CGL and CHSL

?

Circle :¡ú

Sector :-

?¦È=

l ¡ú length of arc AB

r ¡ú radius

¡ý

always in Radian

180¡ã

IC =

¦Ð

C

¦Ð = 180¡ã

I¡ã =

¦ÐC

180

¦È

? Length of arc = 2¦Ðr 360¡ã

¦È

360¡ã

¦È

? Perimeter of sector = ¦Ðr

+ 2r

180¡ã

? Area of sector OAB = ¦Ðr 2

Segment :-

¡ú Area of segment = area of sector OACB ¨C area of ?OAB

¦È

1

= ¦Ðr 2 360¡ã ? 2 r 2 sin ¦È

¡ú Perimeter = length of arc ACB + Chord length AB

¦È

¦È

= (2¦Ðr) 360¡ã + 2r sin (2)

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Q1. Find the area of a segment of a circle with a central angle of 120 degrees and a radius of 8 cm.

Sol. Area of segment = ¦Ðr 2

=

120¡ã

¦Ð(8)2 360¡ã

= 83.047

1

¦È

360¡ã

1

? r 2 sin ¦È

2

2

? 2 (8) 120¡ã

Q2. Find the area of a sector with an arc length of 30 cm and a radius of 10 cm.

¦È

Sol. Length of arc = 2¦Ðr 360¡ã =30

¦È

¦Ðr 360¡ã =15

¦È

¦È

Area of sector OAB = ¦Ðr 2 360¡ã = (¦Ðr 360¡ã) r = 15 ¡Á 10 = 150 cm

Q3. In a circle of radius 21 cm and arc subtends an angle of 72 at centre. The length of arc is?

Sol. Length of arc = 2¦Ðr

= 2 ¦Ð ¡Á 21 ¡Á

72¡ã

360¡ã

¦È

360¡ã

= 26.4 cm

Important Properties Of Circle : ?

Perpendicular from the centre of a circle to a chord bisects the chord.

AM = MB

Q1. AB = 8 cm and CD = 6 cm are two parallel chords on the same side of the center of the circle.

The distance between them is 1 cm. Find the length of the radius?

Sol.

Let ON = x , AO = r

In triangle AOE

r2 = 16 + (x-1)2

In triangle OCN

r2 = 9 +x2

16 + (x-1)2 = 9 +x2

x=4

r2 = 9 +16, r = 5 cm

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?

Chords corresponding to equal arcs are equal.

? = CD

? , then chord , AB = CD

If AB

?

Equal Chords of Circle Subtends equal angles at the centre.

If AB = CD

then ¡Ï1 = ¡Ï2

?

Equal chords of a circle are equidistance from the centre.

If AB = CD, Then OX = OY

?

The angle subtended by an arc of a circle at the centre is double the angle subtended by it at any

point on the remaining part of the circle.

x = 2y

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Q1. The length of chord of a circle is equal to the radius of the circle .The angle which this chord

subtends in the major segment of the circle is equal to?

Sol.

OA = OB = r

AB is equal to radius

Therefore triangle OAB is an equilateral triangle

Angle OAB = 60¡ã

Angle ACB, angle which chord subtends at major angle =

?

60¡ã

2

= 30¡ã

Angle in same segment of a circle are equal.

¡Ï1 = ¡Ï2

?

Angle in a semicircle is always a right angle.

Q1. AC is the diameter of a circumcircle of triangle ABC. Chord

ED is parallel to the diameter AC. If Angle CBE = 50¡ã, then

the measure of angle DEC is?

Sol.

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Angle CBE = 50¡ã

Angle ABC = 90¡ã (Angle in a semicircle is always a right angle)

Angle ABE = 90¡ã - 50¡ã = 40¡ã

Angle ABE = Angle ACE = 40¡ã

Angle ACE = Angle CED = 40¡ã (Alternate Angles)

?

If, ABCD is a cyclic quadrilateral

¡ÏA + ¡ÏC = 180¡ã

¡ÏB + ¡ÏD = 180¡ã

?

ABCD is a cyclic quadrilateral

¡Ï1 = ¡Ï2

?

A tangent at any point of circle is Perpendicular to the radius through the point of contact

OP¡ÍAB

?

PA ¡Á PB = PC ¡Á PD

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