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.
\( \begin{aligned} \require{color}
\angle SDT &= 90^{\circ} &\color{green} \text{angle in semi circle, diameter } AB \\
\angle ASC &= 90^{\circ} &\color{green} \text{angle in semi circle, diameter } AC \\
\angle DSC &= 90^{\circ} &\color{green} AD \text{ is a straight line} \\
\angle CTB &= 90^{\circ} &\color{green} \text{angle in semi circle, diameter } CB \\
\angle CTD &= 90^{\circ} &\color{green} DB \text{ is a straight line} \\
\angle SCT &= 90^{\circ} &\color{green} \text{angle sum of quadrilateral} \\
\therefore CTDS &\text{ is a rectangle} &\color{green} \text{quadrilateral with all angles right angles} \\
\end{aligned} \\ \)
(b)   Show that \( \triangle MXS \) and \( \triangle MXC \) are congruent.
\( \begin{aligned} \require{color}
MX &= MX &\color{green} \text{common side} \\
MS &= MC &\color{green} \text{same radii} \\
SX &= XC &\color{green} \text{diagonals bisect each other} \\
\therefore \triangle MXS &\equiv \triangle MXC &\color{green} \text{SSS} \\
\end{aligned} \\ \)
(c)   Show that the line \( ST \) is a tangent to the semicircle with diameter \( AC \).
\( \begin{aligned} \require{color}
\angle MCX &= 90^{\circ} &\color{green} \text{given} \\
\angle MSX &= \angle MCX &\color{green} \text{corresponding angles of congruent triangles} \\
\angle MCX &= 90^{\circ} \\
\therefore ST \text{ is a } &\text{tangent to the circle} &\color{green} \text{meets radius on the circumference at right angles} \\
\end{aligned} \\ \)

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