Answer:
[tex]\textsf{a)} \quad x = 63^{\circ}[/tex]
[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)
The triangle inside the circle is an isosceles triangle, since two of its sides are the radius of the the circle (and therefore equal in length).
Therefore, its two base angles are 27°.
A tangent is a straight line that touches a circle at only one point.
The tangent of a circle is always perpendicular to the radius.
[tex]\implies x + 27^{\circ} = 90^{\circ}[/tex]
[tex]\implies x +27^{\circ}-27^{\circ}= 90^{\circ} - 27^{\circ}[/tex]
[tex]\implies x = 63^{\circ}[/tex]
Part (b)
The circle theorem that allows the calculation of 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]
