Symmetry Properties of a Circle

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Symmetry Properties of a Circle
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Chords

• A and B are any two points on the circumference
  of the circle , center O.
• The line segment AB is called a chord.
•  What is the longest chord of a circle?

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Symmetry Properties of the chords of a circle

• The line through the midpoint of a chord of a
  circle and the centre of the circle is
  perpendicular to the chord.

If AC BC then ?OCA ?OCB 90?
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Symmetry Properties of the chords of a circle

• The perpendicular from the centre of a circle to
  a chord of the circle bisects the chord.

If ?OCB 90?, then AC BC
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Symmetry Properties of the chords of a circle

• OA OB
• ?OAC ?OBC
• ?OAB is an isosceles triangle

Note ?AOC ?BOC as ?OAC ? ?OBC
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Symmetry Properties of the chords of a circle

• Equal chords of a circle are equidistant from the
  centre of the circle.

If AB CD, then OE OF
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Symmetry Properties of the chords of a circle

• Chords of a circle which are equidistant from the
  centre of the circle are equal in length.

If OE OF, then AB CD
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Time to Work

• Ex 11A Page 77
• Q 1 to 3

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Tangent

• A tangent is a line that intersects the circle at
  exactly one point and is perpendicular to a
  radius at that point of intersection.

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Symmetry Properties of the Tangents to a circle

• A tangent to a circle is perpendicular to the
  radius of the circle at the point of contact.

• OB is the radius.
• ABC is the tangent.

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Symmetry Properties of the Tangents to a circle

• Tangents drawn from an external point to a circle
  are equal in length.

• OA OC (radius)
• AB and BC are the tangents
• Hence AB CB

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Symmetry Properties of the Tangents to a circle

• The angle between the tangents drawn from an
  external point to a circle is bisected by the
  line through the external point of the centre of
  the circle.

• Given AB and CB are the tangents, then
• ?OAB ?OCB 90?
• ?OBA ?OBC
• ?AOB ?COB

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Time to Work

• Class Work
• Skill Practice 11B Page 81
• Q1a,d
• Q2c,d
• Q3c,d
• Q5
• Q7
• Q8

• Home work
• Skill Practice 11B Page 81
• Q1b, c
• Q2a, b
• Q3a, b
• Q4
• Q6

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Property 1

• A circle is symmetrical about every diameter.
  Hence any chord AB perpendicular to a diameter is
  bisected by the diameter.
• Also, any chord bisected by a diameter is
  perpendicular to the diameter.