The circle below is centred at O.
a) Work out the size of angle x.
b) Which of the circle theorems below allows
you to calculate this angle?
O
X
Not drawn accurately
31°
The perpendicular line from the centre of a circle to a chord
bisects the chord
Two tangents that meet at a point are the same length
The angle at the circumference in a semicircle is a right angle
Opposite angles in a cyclic quadrilateral add up to 180°
The angle between the tangent and the radius at a point
on a circle is 90°

[tex]\textsf{b)} \quad \boxed{\begin{minipage}{8.5 cm}\sf The angle between the tangent and the radius at a point \\on a circle is $90^{\circ}$.\end{minipage}}[/tex]

Step-by-step explanation:

Part (a)

As two sides of the triangle inside the circle are the radius of the circle, the triangle is an isosceles triangle.

Therefore, its two base angles are 31°.

The tangent of a circle is:

A straight line that touches the circle at only one point.
Always perpendicular to the radius.

Therefore:

[tex]\implies x + 31^{\circ} = 90^{\circ}[/tex]

[tex]\implies x +31^{\circ}-31^{\circ}= 90^{\circ} - 31^{\circ}[/tex]

[tex]\implies x = 59^{\circ}[/tex]

Part (b)

The circle theorem that allows you to calculate angle x is:

[tex]\boxed{\begin{minipage}{8.5 cm}\sf The angle between the tangent and the radius at a point \\on a circle is $90^{\circ}$.\end{minipage}}[/tex]