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.
You must giv... show full transcript
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 One Pair of Equal Angles
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Answer
Notice that angle ABE is equal to angle DCE, as they subtend the same arc AC in the circle. Thus, we can establish that:
∠ABE=∠DCE
This is a key property of angles inscribed in a circle.
Step 2
Identify Another Pair of Equal Angles
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Answer
Next, we observe the angles at points E and C. Since both angles EAB and ECD intercept arc AD, they are equal:
∠EAB=∠ECD
This shows that we have established two pairs of equal angles.
Step 3
Conclude Similarity of Triangles
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Answer
Having established that:
∠ABE=∠DCE
∠EAB=∠ECD
We can conclude that triangle ABE is similar to triangle DCE by the AA (Angle-Angle) criterion for similarity, as two pairs of corresponding angles are equal:
