Question

A right triangle ABC is inscribed in a circle Γ with center O. Let AC be the diameter of Γ. Let l be the tangent of Γ, being tangent at point B.

Prove that AC is parralel to l​

Answer 1

Verified answer

To prove that AC is parallel to the tangent line l at point B, we need to show that the opposite interior angles are congruent.

Let's consider the triangle ABC inscribed in the circle Γ with center O. Since AC is the diameter of the circle, angle ACB is a right angle (90 degrees).

Now, we can prove that angle CBA is congruent to angle CAB, which will imply that AC is parallel to the tangent line l.

Proof:

1. In triangle ABC, angle CAB and angle ACB form a linear pair, which means they add up to 180 degrees.

2. Since angle ACB is a right angle (90 degrees), the sum of angle CAB and angle CBA is also 90 degrees.

3. In the circle Γ, the measure of an inscribed angle is half the measure of its intercepted arc.

4. The intercepted arc for angle CAB is arc CB, and the intercepted arc for angle CBA is arc CA.

5. Since angle CAB and angle CBA add up to 90 degrees, the intercepted arcs arc CB and arc CA must add up to 180 degrees.

6. If the intercepted arcs arc CB and arc CA add up to 180 degrees, it means they are opposite arcs.

7. Opposite arcs in a circle are congruent.

8. Therefore, arc CB is congruent to arc CA.

9. By the converse of the alternate segment theorem, angles CAB and CBA are congruent.

10. Congruent angles have equal measures, so angle CAB and angle CBA both measure 45 degrees.

11. Since angle CBA measures 45 degrees, it is a right angle.

12. Therefore, angle CBA and angle ACB are congruent.

13. Congruent angles are opposite interior angles of parallel lines.

14. Hence, AC is parallel to the tangent line l.

Therefore, we have proved that AC is parallel to the tangent line l at point B.