Angle at centre questions with answers

Questions

1. Question 1Find the angle at the centre

   In the diagram, $A$, $B$ and $C$ lie on the circle with centre $O$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠AOB$\mathstrut \angle AOB = 2 \times 52^\circ$ (∠ at centre = 2 ∠ at circumference)$x = 104$
2. Question 2Find the angle at the circumference

   In the diagram, $A$, $B$ and $C$ lie on the circle with centre $O$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠ACB$\mathstrut \angle ACB = \dfrac{92^\circ}{2}$ (∠ at centre = 2 ∠ at circumference)$x = 46$
3. Question 3Point on the minor arc (reflex angle)

   In the diagram, $A$, $B$ and $C$ lie on the circle with centre $O$. Find $x$.

   Show the worked solution

   1. Step 1: Which angle at O stands on the same arc as the angle at C?$\mathstrut \text{Reflex } \angle AOB = 360^\circ - 144^\circ$ (∠s at a point)$\text{Reflex } \angle AOB = 216^\circ$
   2. Step 2: Find ∠ACB$\mathstrut \angle ACB = \dfrac{216^\circ}{2}$ (∠ at centre = 2 ∠ at circumference)$x = 108$
4. Question 4With an isosceles triangle

   In the diagram, $A$, $B$ and $C$ lie on the circle with centre $O$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠OBA, then ∠AOB$\angle OBA = 22^\circ$ (base ∠s of isos. △)$\mathstrut \angle AOB = 180^\circ - 2 \times 22^\circ$ (∠ sum of △)$\angle AOB = 136^\circ$
   2. Step 2: Find ∠ACB$\mathstrut \angle ACB = \dfrac{136^\circ}{2}$ (∠ at centre = 2 ∠ at circumference)$x = 68$
5. Question 5Lines that cross (O outside the triangle)

   In the diagram, $A$, $B$ and $C$ lie on the circle with centre $O$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠ACB$\mathstrut \angle ACB = \dfrac{88^\circ}{2}$ (∠ at centre = 2 ∠ at circumference)$x = 44$
6. Question 6One line through the centre

   In the diagram, $A$, $B$ and $C$ lie on the circle with centre $O$. Find $x$.

   Show the worked solution

   1. Step 1: Find ∠ACB$\mathstrut \angle ACB = \dfrac{62^\circ}{2}$ (∠ at centre = 2 ∠ at circumference)$x = 31$