A, B and C are points on the circumference of a circle, centre O - Edexcel - GCSE Maths - Question 20 - 2017 - Paper 3
Question 20
A, B and C are points on the circumference of a circle, centre O. AOB is a diameter of the circle.
Prove that angle ACB is 90°.
You must not use any circle theorem... show full transcript
Worked Solution & Example Answer:A, B and C are points on the circumference of a circle, centre O - Edexcel - GCSE Maths - Question 20 - 2017 - Paper 3
Step 1
Draw triangle ABC and identify relevant angles
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Answer
Begin by drawing triangle ABC with points A, B, and C on the circumference of the circle, and the center O. Note that AOB is a straight line, meaning that the angle AOB is 180°. Since O is the center of the circle, OA = OB (both are radii of the circle).
Step 2
Find the sum of angles in triangle ABC
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Answer
In triangle ABC, the sum of the interior angles can be expressed as:
A+B+C=180°
Here, let angle ACB = C, angle CAB = A, and angle ABC = B. By substituting angle AOB into this equation, we get:
\Rightarrow C + (180° - C) = 180°$$
Step 3
Rearrangement and conclusion
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Answer
Rearranging the earlier expression leads to:
\Rightarrow C = 90°$$
Thus, we conclude that angle ACB measures 90°, proving that it is a right angle.
