Basic Properties of Circles: practice questions

6 questions · 11 marks · free, no account · Read the notes first

1. Question 1Multiple choice · 1 mark

   $O$ is the centre of a circle. $A$, $B$ and $C$ lie on the circle, with $C$ on the major arc $AB$. If $\angle AOB = 130°$, find $\angle ACB$.

   A

   $50°$

   B

   $65°$

   C

   $115°$

   D

   $130°$

2. Question 2Multiple choice · 1 mark

   $ABCD$ is a cyclic quadrilateral with $\angle A = 72°$. Find $\angle C$.

   A

   $72°$

   B

   $98°$

   C

   $108°$

   D

   $118°$

3. Question 3Multiple choice · 1 mark

   A chord of length 16 cm is 6 cm from the centre of a circle. Find the radius of the circle.

   A

   8 cm

   B

   12 cm

   C

   14 cm

   D

   10 cm

4. Question 4Multiple choice · 1 mark

   $AB$ is a diameter of a circle and $C$ is a point on the circle. If $\angle CAB = 35°$, find $\angle CBA$.

   A

   $55°$

   B

   $35°$

   C

   $65°$

   D

   $45°$

5. Question 54 marks

   $TA$ and $TB$ are tangents to a circle with centre $O$, touching it at $A$ and $B$. If $\angle ATB = 50°$, find $\angle AOB$ and $\angle OAB$, giving reasons.

   Show worked solution

   $\angle OAT = \angle OBT = 90°$ (tangent ⊥ radius) 1A

   $\angle AOB = 360° - 90° - 90° - 50° = 130°$ (∠ sum of polygon) 1A

   $OA = OB$ (radii), so $\angle OAB = \angle OBA$ (base ∠s, isos. △) 1M

   $\angle OAB = \dfrac{180° - 130°}{2} = 25°$ (∠ sum of △) 1A

6. Question 63 marks

   $TA$ is the tangent to a circle at $A$. $AB$ is a chord and $C$ is a point on the circle on the opposite side of $AB$ from $T$. If $\angle TAB = 68°$ and $\angle CAB = 47°$, find $\angle ACB$ and $\angle ABC$, giving reasons.

   Show worked solution

   $\angle ACB = \angle TAB = 68°$ 1A

   (∠ in alt. segment) 1A

   $\angle ABC = 180° - 68° - 47° = 65°$ (∠ sum of △) 1A