GCSE OCR Maths Circle theorems: A, B, C and D are points on the

A, B, C and D are points on the circumference of a circle, centre O - OCR - GCSE Maths - Question 8 - 2018 - Paper 1

Question 8

A, B, C and D are points on the circumference of a circle, centre O. Angle CAD = 28° and CD = 6.4 cm. BD is a diameter of the circle. Calculate the area of the cir... show full transcript

Worked Solution & Example Answer:A, B, C and D are points on the circumference of a circle, centre O - OCR - GCSE Maths - Question 8 - 2018 - Paper 1

Step 1

Find angle CBD

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Answer

Since BD is a diameter, angle BCD is a right angle (90°). Therefore, we can use the triangle properties to find angle CBD:

$ext{angle CBD} = 180° - ext{angle CAD} - 90° = 180° - 28° - 90° = 62°$

Step 2

Use the sine rule to find BD

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Answer

Using the sine rule in triangle BCD:

$\frac{CD}{\sin(\text{angle CBD})} = \frac{BD}{\sin(\text{angle BCD})}$

Substituting the known values:

$\frac{6.4}{\sin(62°)} = \frac{BD}{\sin(90°)}$

Thus,

$BD = 6.4 \times \sin(90°) / \sin(62°)$

Calculating gives

$BD = 6.4 / \sin(62°) \approx 7.75 cm$

Step 3

Find the radius of the circle

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Answer

Since BD is the diameter, the radius (r) can be determined as follows:

$r = \frac{BD}{2} = \frac{7.75}{2} \approx 3.875 cm$

Step 4

Calculate the area of the circle

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Answer

The area (A) of the circle is given by the formula:

$A = \pi r^2$

Substituting the radius we found:

$A = \pi (3.875)^2 \approx \pi (15.015625) \approx 47.12 cm^2$

A, B, C and D are points on the circumference of a circle, centre O - OCR - GCSE Maths - Question 8 - 2018 - Paper 1

Worked Solution & Example Answer:A, B, C and D are points on the circumference of a circle, centre O - OCR - GCSE Maths - Question 8 - 2018 - Paper 1

Find angle CBD

Use the sine rule to find BD

Find the radius of the circle

Calculate the area of the circle

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