# 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 $$AC$$.