Circle Geometry with Semicircles

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 \( AC \).

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

Unlock your full learning potential—download our expertly crafted slide files for free and transform your self-study sessions!

Discover more enlightening videos by visiting our YouTube channel!

 

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume




Related Articles

Responses

Your email address will not be published. Required fields are marked *