The diagram shows a circle, centre O. The points M, N & P lie on circumference of circle. AMB is...

This video explains an example from "Circle Theorem".

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(a)
The diagram shows a circle, centre O.
The points M, N & P lie on circumference of the circle.
AMB is a tangent to the circle at M.

Find the value of f and the value of g.

(b)
The diagram shows a cyclic quadrilateral.
Find the value of k.

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Note : The question is referred from CIE IGCSE MAY / JUNE 2024 Paper 4 (Extended), 0580/42/M/J/24, 0580 s24 qp 42, for educational purpose only.

[Grizzly-vrpn]

Question 1:
I solved it in the opposite 'direction':
OP = OM, so △POM is isosceles, therefore ∠OMP = ∠OPM = g
Using tangent to a circle theorem ∠OMB = 90°, so g and ∠PMB (= 56°) are complimentary, so g = 90° – 56° = 34°.
So ∠MOP = 180° – (2 x 34)° = 112°
Using inscribed angle theorem f = (∠MOP)/2 = 56°