A and B are points on the circumference of a circle, centre O - OCR - GCSE Maths - Question 13 - 2017 - Paper 1
Question 13
A and B are points on the circumference of a circle, centre O.
CA and CB are tangents to the circle.
Prove that triangle OAC is congruent to triangle OBC.
Worked Solution & Example Answer:A and B are points on the circumference of a circle, centre O - OCR - GCSE Maths - Question 13 - 2017 - Paper 1
Step 1
OC is common or shared
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Answer
In triangle OAC and triangle OBC, segment OC is the same for both triangles. This satisfies one pair of equal sides.
Step 2
OA = OB (equal radii)
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Answer
The radii OA and OB are equal since they are both radii of the same circle with center O.
Step 3
∠AOC = ∠BOC (equal angles)
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Answer
The angles ∠AOC and ∠BOC are equal because CA and CB are tangents to the circle from point C, and the angles formed with the line segments OA and OB are equal as a result.
Step 4
CA = CB (tangents from a point)
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Answer
By the property of tangents, the lengths CA and CB are equal since they are both tangents drawn from point C to the circle.
Step 5
Conclusion
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Answer
Thus, by the Side-Angle-Side (SAS) criterion for triangle congruence, triangle OAC is congruent to triangle OBC.
