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 centre’s equal angles at the centre stand on equal chords
• the angles at the centre are twice an angle at the circumference subtended at 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 on an Example of Circle Geometry with Semi Circles

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

\begin{aligned} \require{AMSsymbols} \require{color} \angle SDT &= 90^{\circ} &\color{green} \text{angle in semi circle, diameter } AB \\ \angle ASC &= 90^{\circ} &\color{green} \text{angle in semi circle, diameter } AC \\ \angle DSC &= 90^{\circ} &\color{green} AD \text{ is a straight line} \\ \angle CTB &= 90^{\circ} &\color{green} \text{angle in semi circle, diameter } CB \\ \angle CTD &= 90^{\circ} &\color{green} DB \text{ is a straight line} \\ \angle SCT &= 90^{\circ} &\color{green} \text{angle sum of quadrilateral} \\ \therefore CTDS &\text{ is a rectangle} &\color{green} \text{quadrilateral with all angles right angles} \end{aligned}

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

\begin{aligned} \require{AMSsymbols} \require{color} MX &= MX &\color{green} \text{common side} \\ MS &= MC &\color{green} \text{same radii} \\ SX &= XC &\color{green} \text{diagonals bisect each other} \\ \therefore \triangle MXS &\equiv \triangle MXC &\color{green} \text{SSS} \end{aligned}

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

\begin{aligned} \require{AMSsymbols} \require{color} \angle MCX &= 90^{\circ} &\color{green} \text{given} \\ \angle MSX &= \angle MCX &\color{green} \text{corresponding angles of congruent triangles} \\ \angle MCX &= 90^{\circ} \\ \therefore ST \text{ is a } &\text{tangent to the circle} &\color{green} \text{meets radius on the circumference at right angles} \end{aligned}