Circle Geometry with Semicircles

There are many properties of circle geometry with semi circles, such as

  • equal arcs on circles of equal radii subtend equal angles at the centres equal angles at the centre stand on equal chords
  • the angles at the centre is twice an angle at the circumference subtended ay the same arc
  • the perpendicular from the centre of a circle to a chord bisects the chord
  • equal chords in equal circles are equidistant from the centres
  • angles in the same segment are equal
  • the angle in a semi-circle is a right angle
  • opposite angles of a cyclic quadrilateral are supplementary

Worked Example of Circle Geometry with Semi Circles

(a)    Explain why \( CTDS \) is a rectangle.

(b)    Show that \( \triangle MXS \) and \( \triangle MXC \) are congruent.

(c)    Show that the line \( ST \) is a tangent to the semicircle with diameter \( AC \).

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