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ABDE is a cyclic quadrilateral - Edexcel - GCSE Maths - Question 16 - 2022 - Paper 3
Question 16
ABDE is a cyclic quadrilateral. ABC and EDC are straight lines. Angle DBC = 60° Given that size of angle EAB: size of angle BCD = 2:1 work out the size of angle BC... show full transcript
Worked Solution & Example Answer:ABDE is a cyclic quadrilateral - Edexcel - GCSE Maths - Question 16 - 2022 - Paper 3
Step 1
size of angle EAB: size of angle BCD = 2:1
Answer
Let the size of angle BCD be denoted as x. Then, according to the ratio given, the size of angle EAB would be 2x. Since ABDE is a cyclic quadrilateral, we can apply the angle properties of cyclic figures.
Using the cyclic quadrilateral property: Substituting the angles we have:
Now, we need to find \angle ADC. Since ABC and EDC are straight lines, we know: Thus, we can express \angle ADC as:
Substituting this back into the cyclic quadrilateral equation:
the equation simplifies to: Now, isolating x gives us: Hence, x = 0° / 3 = 0°
This means that no such angle BCD can exist based on the imposed ratios as it contradicts the earlier conditions.
Step 2
Calculate angle BCD
Answer
Since our initial assumption leads to a contradiction, let’s set the angle EAB as 2y:
Thus, according to the cyclic property, we can set up: With angle ADC still being 120° as above: Solving gives us: .
Therefore, the workable answer should consider valid definitions and angles, suggesting that the cyclic properties may lead to contradictory results depending on the selection of alternate angles for further verification. The solving approach must validate consistent cyclic angle rules.