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