A, B, C and D are four points on the circumference of a circle - Edexcel - GCSE Maths - Question 15 - 2017 - Paper 2
Question 15
A, B, C and D are four points on the circumference of a circle.
AEC and BED are straight lines.
Prove that triangle ABE and triangle DCE are similar.
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Worked Solution & Example Answer:A, B, C and D are four points on the circumference of a circle - Edexcel - GCSE Maths - Question 15 - 2017 - Paper 2
Step 1
Identify equal angles (angle ABE = angle DCE)
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Answer
Since A, B, C, and D are points on the circumference of the circle, the angles at the circumference subtended by the same arc are equal. Here, angle ABE is subtended by arc AE, and angle DCE is subtended by arc DE. Therefore, we have:
extangleABE=extangleDCE
Step 2
Identify another pair of equal angles (angle EAB = angle EDC)
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Answer
Similarly, angles EAB and EDC are subtended by the same arc AC, making them equal:
extangleEAB=extangleEDC
Step 3
Conclude similarity of triangles (triangle ABE ~ triangle DCE)
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Answer
With two pairs of equal angles identified, we can conclude that triangle ABE is similar to triangle DCE by the Angle-Angle (AA) criterion for similarity of triangles:
