In the diagram above, O is the centre of the circle. Use the diagram above to prove the theorem which states that the angle subtended by a chord at the centre of the circle is equal to twice the angle subtended by the same chord at the circumference, that is, prove that $T\widehat{O}P = 2T\widehat{K}P$.

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In the diagram above, O is the centre of the circle - NSC Mathematics - Question 8 - 2023 - Paper 2

Question 8

In the diagram above, O is the centre of the circle. Use the diagram above to prove the theorem which states that the angle subtended by a chord at the centre of th... show full transcript

Worked Solution & Example Answer:In the diagram above, O is the centre of the circle - NSC Mathematics - Question 8 - 2023 - Paper 2

Step 1

Prove that $T\widehat{O}P = 2T\widehat{K}P$

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Answer

Draw line segment $KO$ to create triangles.
Note that angles $T\widehat{K}O$ and $K\widehat{O}T$ are equal. Consequently, angle $T\widehat{O}P$ is formed at the centre.
Recognize that $\angle KTP = 180^\circ - 2\widehat{K}O$ (sum of angles in triangle).
For the angle at point $P$ , since angles in a triangle sum to $180^\circ$ , conclude that: $T\widehat{O}P = 2T\widehat{K}P$ .

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