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A, B, R and P are four points on a circle with centre O - Edexcel - GCSE Maths - Question 1 - 2018 - Paper 3

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A, B, R and P are four points on a circle with centre O. A, O, R and C are four points on a different circle. The two circles intersect at the points A and R. CPA, C... show full transcript

Worked Solution & Example Answer:A, B, R and P are four points on a circle with centre O - Edexcel - GCSE Maths - Question 1 - 2018 - Paper 3

Step 1

Prove that angle CAB = 180 - angle OAR

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Answer

Using the fact that angles subtended by the same chord at the circumference of a circle are equal, we can write:

∠CAB+∠OAR=180∘\angle CAB + \angle OAR = 180^{\circ}

This is because CAB and OAR subtend the same arc AB.

Step 2

Establish relationship using angle on a straight line

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Answer

Since A, O, R are collinear, we have:

∠OAR+∠CRB=180∘\angle OAR + \angle CRB = 180^{\circ}

This implies that:

∠CAB=180∘−∠CRB.\angle CAB = 180^{\circ} - \angle CRB.

Step 3

Use the properties of circles

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Answer

The angles subtended by the same arc are equal, which tells us:

∠ABC=∠CRB.\angle ABC = \angle CRB.

Step 4

Complete proof and conclusion

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Answer

From the previous relationships, we have:

∠CAB=180∘−∠CRB\angle CAB = 180^{\circ} - \angle CRB and \angle ABC = \angle CRB.$$ Therefore,

∠CAB=∠ABC.\angle CAB = \angle ABC. Hence proved.

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