Question

Prove that in this diagram where O is the centre of a unit circle, the triangles POT and QOR are congruent.

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Answer 1

Here are the steps to prove that triangles POT and QOR are congruent:

1) The circle has center O and radius 1, so OP = OQ = 1 (radius of the circle)

2) Angles POQ and ROQ are inscribed in the same arc PQ, so they are congruent (inscribed angle theorem)

3) Triangle POT is isosceles with OP = PT

4) Triangle QOR is isosceles with OQ = QR

5) By Side-Angle-Side postulate (SAS), triangles POT and QOR are congruent since:

a) OP = OQ (both radii are 1)

b)∠POQ ≅ ∠ROQ (inscribed angles are congruent)

c) PT = QR (corresponding sides of isosceles triangles)

In summary, triangles POT and QOR are congruent by the Side-Angle-Side congruence postulate since they have 2 matching sides (OP and OQ) and a matching acute angle (POQ and ROQ).