Investigate the pattern in the circle. △ABC

Question

is a right triangle inscribed in circle R
such that chord AB
is a diameter. Find the measure of ∠ABC
if arc BC
is 62 degrees.

(1 point)
Responses

∠ABC
is 31 degrees.
triangle upper A upper B upper C is 31 degrees.

∠ABC
is 59 degrees.
triangle upper A upper B upper C is 59 degrees.

∠ABC
is 90 degrees.
triangle upper A upper B upper C is 90 degrees.

∠ABC
is 118 degrees.

Answer 1

Since triangle \( ABC \) is a right triangle inscribed in circle \( R \) with chord \( AB \) as a diameter, we know that the angle \( \angle ACB \) (the angle opposite the diameter) must be \( 90^\circ \) according to the Inscribed Angle Theorem.

Given that arc \( BC \) measures \( 62^\circ \), we can determine \( \angle ABC \) as follows:

1. The angle that intercepts the arc \( BC \) is \( \angle BAC \). According to the Inscribed Angle Theorem, the measure of the angle is half the measure of the arc that it intercepts. Therefore:

\[ \angle BAC = \frac{1}{2} \times \text{arc } BC = \frac{1}{2} \times 62^\circ = 31^\circ \]

2. Now, since \( \triangle ABC \) is a right triangle:

\[ \angle ABC + \angle ACB + \angle BAC = 180^\circ \]

Given \( \angle ACB = 90^\circ \) and \( \angle BAC = 31^\circ \):

\[ \angle ABC + 90^\circ + 31^\circ = 180^\circ \]

\[ \angle ABC + 121^\circ = 180^\circ \]

\[ \angle ABC = 180^\circ - 121^\circ = 59^\circ \]

Thus, the measure of \( \angle ABC \) is \( 59^\circ \).

The correct response is:

∠ABC is 59 degrees.