If a right triangle is inscribed in a circle, then the hypotenuse is a diameter. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.
The theorem is written as a conditional statement and its converse.
Conditional: If a right triangle inscribed in a circle, then the hypotenuse is a diameter.
Converse: If one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle.
We will write a plan for each statement. Let's start with the conditional statement!
Conditional Statement
We can visualize the conditional statement as follows.
Given that △ ABC is a right triangle, we want to prove that AC is a diameter. To do so, we can follow the following steps.
Use the definition of a semicircle to show that AC is a semicircle and AC is a diameter.
Using this plan, we can prove the conditional statement. Next, we will write a plan for the converse of the conditional statement.
Converse of the Conditional Statement
Let's first visualize the statement!
By the converse statement, we know that AC is a diameter of ⊙ O and it is the hypotenuse of △ ABC. With this information, we want to prove that △ ABC is a right triangle with m∠ ABC = 90^(∘). Therefore, we can write the plan as follows.
Use the definition of a semicircle to show that AC is a semicircle.
Find the measure of AC.
By the Intercepted Angle Theorem, find the measure of ∠ ABC.
With this plan, we can prove the converse of the conditional statement.
