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Summary

8.3 Summary (EMBJC)

  • Arc An arc is a portion of the circumference of a circle.
  • Chord - a straight line joining the ends of an arc.
  • Circumference - perimeter or boundary line of a circle.
  • Radius (\(r\)) - any straight line from the centre of the circle to a point on the circumference.
  • Diameter - a special chord that passes through the centre of the circle. A diameter is the length of a straight line segment from one point on the circumference to another point on the circumference, that passes through the centre of the circle.
  • Segment A segment is a part of the circle that is cut off by a chord. A chord divides a circle into two segments.
  • Tangent - a straight line that makes contact with a circle at only one point on the circumference.
  • A tangent line is perpendicular to the radius, drawn at the point of contact with the circle.

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  • If \(O\) is the centre and \(OM \perp AB\), then \(AM = MB\).
  • If \(O\) is the centre and \(AM = MB\), then \(A\hat{M}O = B\hat{M}O = \text{90}\text{°}\).
  • If \(AM = MB\) and \(OM \perp AB\), then \(\Rightarrow MO\) passes through centre \(O\).
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If an arc subtends an angle at the centre of a circle and at the circumference, then the angle at the centre is twice the size of the angle at the circumference.

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Angles at the circumference subtended by the same arc (or arcs of equal length) are equal.

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The four sides of a cyclic quadrilateral \(ABCD\) are chords of the circle with centre \(O\).

  • \(\hat{A} + \hat{C} = \text{180}\text{°}\) (opp. \(\angle\)s supp.)
  • \(\hat{B} + \hat{D} = \text{180}\text{°}\) (opp. \(\angle\)s supp.)
  • \(E\hat{B}C = \hat{D}\) (ext. \(\angle\) cyclic quad.)
  • \(\hat{A}_1 = \hat{A}_2 = \hat{C}\) (vert. opp. \(\angle\), ext. \(\angle\) cyclic quad.)
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Proving a quadrilateral is cyclic: If \(\hat{A} + \hat{C} = \text{180}\text{°}\) or \(\hat{B} + \hat{D} = \text{180}\text{°}\), then \(ABCD\) is a cyclic quadrilateral.

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If \(\hat{A}_1 = \hat{C}\) or \(\hat{D}_1 = \hat{B}\), then \(ABCD\) is a cyclic quadrilateral.

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If \(\hat{A} = \hat{B}\) or \(\hat{C} = \hat{D}\), then \(ABCD\) is a cyclic quadrilateral.

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If \(AT\) and \(BT\) are tangents to circle \(O\), then

  • \(OA \perp AT\) (tangent \(\perp\) radius)
  • \(OB \perp BT\) (tangent \(\perp\) radius)
  • \(TA = TB\) (tangents from same point equal)
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  • If \(DC\) is a tangent, then \(D\hat{T}A = T\hat{B}A\) and \(C\hat{T}B = T\hat{A}B\)
  • If \(D\hat{T}A = T\hat{B}A\) or \(C\hat{T}B = T\hat{A}B\), then \(DC\) is a tangent touching at \(T\)