Circle Theorems: Question 3

Worked solution

Identifying the correct theorem

The line $PAQ$ touches the circle at $A$, so it is a tangent, and $AB$ is a chord drawn from the point of contact. The angle between the tangent and the chord is $\angle PAB = 64^{\circ}$.

The alternate segment theorem states:

The angle between a tangent and a chord equals the angle in the alternate segment: the angle subtended by that chord from the arc on the other side of the chord.

Here the chord is $AB$ and the angle in the alternate segment is the angle subtended by $AB$ at $C$, namely $\angle ACB$.

Applying the theorem

$\angle ACB = \angle PAB = 64^{\circ}.$

Alternative method (radius, isosceles triangle, angle at centre)

This confirms the answer using two other circle facts.

Let $O$ be the centre and draw radii $OA$ and $OB$.

Step 1: Tangent meets radius at $90^{\circ}$. Since $OA \perp$ tangent, $\angle OAB = 90^{\circ} - \angle PAB = 90^{\circ} - 64^{\circ} = 26^{\circ}.$

Step 2: Isosceles triangle. $OA = OB$ (both radii), so triangle $OAB$ is isosceles and $\angle OBA = \angle OAB = 26^{\circ}.$

Step 3: Angle sum. $\angle AOB = 180^{\circ} - 26^{\circ} - 26^{\circ} = 128^{\circ}.$

Step 4: Angle at centre $= 2\times$ angle at circumference. The chord $AB$ subtends $\angle AOB$ at the centre and $\angle ACB$ at the circumference (with $C$ on the major arc), so $\angle ACB = \tfrac{1}{2}\,\angle AOB = \tfrac{1}{2}(128^{\circ}) = 64^{\circ}.$

Both methods agree.

Answer

$\boxed{\angle ACB = 64^{\circ}} \quad\text{(option B)}$

Note how the figures $26^{\circ}$, $128^{\circ}$ and $116^{\circ}$ all appear as intermediate or mis-paired values, which is exactly why they are tempting wrong answers.