Shape and Space 6
4 marks · HardExplain how the alternate segment theorem can be used to find an unknown angle in a circle. Give a clear step-by-step example, including both the reasoning and the calculation.
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The alternate segment theorem states that the angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment. Example and stepwise reasoning required.
Must show: States the theorem, Describes how to use it, Gives an example with calculation and steps
Common mistakes: Not stating the theorem; No clear example; Not justifying the steps
Show mark scheme and worked solution
Marking points
B1 · 1 mark · Correctly states the alternate segment theorem.
M1 · 1 mark · Describes how the theorem can be used to find an unknown angle.
M2 · 1 mark · Provides an explicit example (with correct working). (follow-through)
A1 · 1 mark · Gives a clear, justified calculation using the theorem. (follow-through)
Worked solution
The alternate segment theorem states: The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment.
For example, suppose a tangent at A touches the circle, and chord AB is drawn. If the angle between the tangent at A and chord AB is 52°, then the angle in the alternate segment, subtended by AB, is also 52° due to the alternate segment theorem.
Step-by-step:
1. Identify the tangent and the chord at the point of contact.
2. Find the angle between the tangent and the chord.
3. The angle in the alternate segment, opposite the chord, is equal to this angle by the theorem.
So if angle TAB = 52°, then angle in the alternate segment subtended by AB = 52°.