Therefore, BA is perpendicular to OA and BC is perpendicular to OC.
BA⊥ OA and BC⊥ OC
Since ∠ OAB and ∠ BCO are right angles, by the definition of a right triangle, we can conclude that △ BAO and △ BCO are right triangles.
△ BAO and △ BCO
are right triangles.
Next, recall that radii of circles are congruent.
From here, we can say that OA is congruent to OC.
OA ≅ OC
See that △ BAO and △ BCO share the same hypotenuse BO. By the Reflexive Property of Congruence, BO is congruent to itself.
BO ≅ BO
Combining all of this information, we can see that the hypotenuse and one leg of △ BAO are congruent to the hypotenuse and one leg of △ BCO.
Thus, by the Hypotenuse-Leg (HL) Theorem, △ BOA is congruent to △ BCO.
△ BAO ≅ △ BCO
Finally, because the corresponding parts of congruent triangles are congruent (CPCTC), we can conclude that BA≅ BC. Let's summarize the above process in a flow proof.
Completed Proof
2
&Given:&& BA and BC are tangent to
& &&⊙ O at A and C, respectively.
&Prove:&& BA ≅ BC
Proof:
