A, B and C are points on a circle of radius 5 cm, centre O - Edexcel - GCSE Maths - Question 18 - 2017 - Paper 3
Question 18
A, B and C are points on a circle of radius 5 cm, centre O.
DA and DC are tangents to the circle.
DO = 9 cm
Work out the length of arc ABC.
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Worked Solution & Example Answer:A, B and C are points on a circle of radius 5 cm, centre O - Edexcel - GCSE Maths - Question 18 - 2017 - Paper 3
Step 1
Recognizing that ∠DAB or ∠DCB is 90° or right angle
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Answer
Since DA and DC are tangents to the circle from point D, we know that the angles ∠DAB and ∠DCB are right angles (90°).
Step 2
Using trigonometry to set up an equation in triangle ODA
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Answer
In triangle ODA, we can use the cosine function to find the length of OD:
extcos(heta)=ODOA
where OA is the radius (5 cm) and OD is given as 9 cm. Hence,
cos(heta)=95.
Step 3
Calculating angle ∠AOD
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Answer
Using the inverse cosine function, we find
θ=cos−1(95)≈0.6435 radians.
Step 4
Finding the length of arc ABC
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Answer
The length of arc ABC can be calculated using the formula:
Length of arc=r×θ
where r is the radius (5 cm). Thus,
Length of arc=5×0.6435≈3.2175 cm.
Step 5
Final Answer
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Answer
Rounding to three significant figures, the length of arc ABC is approximately 3.22 cm.
