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Points B, D, E and F lie on a circle - Edexcel - GCSE Maths - Question 16 - 2020 - Paper 3

Question 16

Points B, D, E and F lie on a circle. ABC is the tangent to the circle at B. Find the size of angle ABD. You must give a reason for each stage of your working.

Worked Solution & Example Answer:Points B, D, E and F lie on a circle - Edexcel - GCSE Maths - Question 16 - 2020 - Paper 3

Step 1

Find angle DFE

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Answer

Since angle DFE is an angle in a cyclic quadrilateral (BDFE), we can use the property that opposite angles in cyclic quadrilaterals add up to 180 degrees. Therefore, we have:

$ext{angle DFE} = 180 - 100 = 80$

Step 2

Find angle EBF

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Answer

Next, we know that angle EBF, being an angle subtended by arc EF at point B, is equal to angle DFE. Thus, we can write:

$ext{angle EBF} = 80$

Step 3

Find angle ABD

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Answer

Finally, angle ABD is the exterior angle at point B, formed by line AB and the chord EF. Exterior angles are equal to the sum of the opposite interior angles. Therefore, we have:

$ext{angle ABD} = ext{angle EBF} + ext{angle DFE} = 80 + 80 = 160$

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