Circles

1. Subject Mathematics
Class IX
•TOPIC
CIRCLES
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2. Angle subtended by a chord at centre of
a circle
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3. Equal chords of a circle subtends equal
angle at centre
• OA = OD (WHY ?)
• OB = OC (WHY ?)
• AB = CD (WHY ?)
• ∆AOB ≅ ∆COD (SSS)
• ∠AOB = ∠COD
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O
BC
D
A

4. If angles subtended by chords of a circle
at centre are equal then the chords are
equal
• OA = OD (WHY ?)
• OB = OC (WHY ?)
• ∠AOB = ∠COD (WHY
?)
• ∆AOB≅ ∆COD (SAS)
• AB = CD
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O
BC
D
A

5. The perpendicular from centre of a
circle to a chord bisects the chord
• OA=OB (WHY ?)
• ∠ OCA=∠OCB
(WHY ?)
• OC=OC (WHY ?)
• ∆ OAC ∆ ≅OBC
(WHY ?)
• AC=BC
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O
A
B
C

6. Line segment joining centre to mid
point of a chord is perpendicular to the
chord
• OA=OB (WHY ?)
• AC=BC (WHY ?)
• OC=OC (WHY ?)
• ∆OAC≅ ∆OBC
(WHY ?)
• ∠ACO= ∠BCO=90
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O
A
B
C

7. Circle through three points
• OA=OB
• OB=OC
• OA=OB=OC
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A
B
C
O

8. SUMMARY
• Equal chords subtend equal angle at centre
• If angle subtended by chords at centre are equal
then chords are equal
• Perpendicular from centre bisects the chord
• Line segment joining mid point of a chord and
centre of circle is perpendicular to the chord
• A circle can be drawn through any three Non
collinear points
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9. • If AB = CD
• Then
• ∠AOB = ∠COD
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O
BC
D
A

10. • If ∠AOB = ∠COD
• Then
• AB = CD
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O
BC
D
A

11. • If OC ⊥ AB
• Then
• AC=BC
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O
A
B
C

12. • IF AC=BC
• THEN
• OC ⊥ AB
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O
A
B
C

13. • Q. From an externl
point P, a tangent PT
and a line segment
PAB is drawn to a
circle with centre O.
ON is perpendicular
on the chord AB.
• Prove that :
• PA.PB = PT2
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14. • Q. If a circle touches
the side BC of a
triangle ABC at P and
extended sides AB
and AC at Q and
R.Prove that
• AQ = ½(BC+CA+AB)
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15. • Q. If a,b,c are the
sides of a right
triangle where c is
the hypotenuse
prove that the radius
r of the circle which
touches the sides of
the triangle is given
by r = (a+b-c)/2
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16. • Q. Angle in alternate
segment.
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17. • Q. AB is a diameter of a circle and chord CD = radius OC. If AC
and BD when produced meet at P prove that angle APB = 60
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18. • Q. Prove that altitudes of a triangle are concurrent.
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