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A and B are points on the circumference of a circle, centre O - OCR - GCSE Maths - Question 13 - 2017 - Paper 1

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Question 13

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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.

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