Use the image to answer the question.

Question

Four labeled points and three line segments connecting the points are plotted on a circle.

Investigate the pattern in the circle. △ABC
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 which answer is correct and why

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

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

∠ABC
is 118 degrees.
triangle upper A upper B upper C is 118 degrees.

∠ABC
is 31 degrees.
triangle upper A upper B upper C is 31 degrees.
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Answer 1

To find the measure of ∠ABC given that arc BC is 62 degrees, we can use the property of angles subtended by an arc in a circle.

When a triangle is inscribed in a circle (circumscribed by the circle), the angle subtended at the circumference (∠ABC in this case) is half the measure of the arc that it subtends.

Since arc BC is given as 62 degrees, we can calculate the measure of ∠ABC as follows:

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

Therefore, the correct answer is:

∠ABC is 31 degrees. triangle ABC is 31 degrees.

This is because angles inscribed in a circle that subtend the same arc are half the measure of the arc itself, and in this case, arc BC measures 62 degrees, leading to ∠ABC measuring 31 degrees.