Year 10A Circle Theorems Worksheets & Practice

Worked examples

Year 10A circle theorems: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

1. Example 1 · Angle at the centre and angle at the circumference

   Points $A$, $B$ and $C$ lie on a circle with centre $O$. The angle at the centre is $\angle AOC = 124°$. Find the angle at the circumference $\angle ABC$.

   Show worked solution

   Key idea: The angle at the centre is twice the angle at the circumference standing on the same arc.

   1. $\angle ABC$ and $\angle AOC$ both stand on arc $AC$.
   2. $\angle ABC = \frac{\angle AOC}{2} = \frac{124°}{2} = 62°$

   Answer: $\angle ABC = 62°$

2. Example 2 · Cyclic quadrilateral: opposite angles

   Points $A$, $B$, $C$ and $D$ lie on a circle in order, forming cyclic quadrilateral $ABCD$. Given that $\angle DAB = 73°$ and $\angle ABC = 105°$, find $\angle BCD$ and $\angle CDA$.

   Show worked solution

   Key idea: Opposite angles of a cyclic quadrilateral sum to $180°$.

   1. $\angle DAB$ and $\angle BCD$ are opposite angles, so:
      $\angle BCD = 180° - 73° = 107°$
   2. $\angle ABC$ and $\angle CDA$ are opposite angles, so:
      $\angle CDA = 180° - 105° = 75°$
   3. Check: $73° + 107° = 180°$ ✓ and $105° + 75° = 180°$ ✓

   Answer: $\angle BCD = 107°$, $\angle CDA = 75°$

3. Example 3 · Tangent and radius at the point of contact

   A straight fence is built so that it just touches a circular pond at a single point $P$. The centre of the pond is $O$ and the radius is $8$ metres. A post is placed at point $T$ on the fence, $15$ metres from $P$. Find the distance from the post at $T$ to the centre $O$.

   Show worked solution

   Key idea: A tangent to a circle is perpendicular to the radius at the point of contact, so $OP \perp TP$ and $\angle OPT = 90°$.

   1. Triangle $OPT$ is right-angled at $P$.
   2. $OP = 8$ m (radius), $TP = 15$ m.
   3. By Pythagoras' theorem:
      $OT^2 = OP^2 + TP^2 = 8^2 + 15^2 = 64 + 225 = 289$
   4. $OT = \sqrt{289} = 17 \text{ m}$

   Answer: The distance from the post to the centre is $17$ metres.

Common mistakes

Where students usually lose marks

Try one yourself

Your turn

Got a different circle theorems question wrong? Upload it to Tuterly and get the worked solution and a fresh one to practise, free.