A

Question

A
C
R
B
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

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

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

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

∠ABC
 is 90 degrees.

GPT-4o mini · Answer

Given that triangle $ \triangle ABC $ is a right triangle inscribed in circle $ R $ with chord $ AB $ as a diameter, we can determine the measure of angle $ \angle ABC $ based on the properties of circles and inscribed angles. 1. Since $ AB $ is a diameter of the circle, it follows that $ \triangle ABC $ is a right triangle with $ \angle ACB $ being the right angle (90 degrees). 2. The measure of the angle $ \angle ABC $ relates to the intercepted arc $ BC $. The measure of the angle formed by two points on the circle (in this case, points $ B $ and $ C $) is half the measure of the arc that they intercept. This is known as the inscribed angle theorem. 3. We are given that arc $ BC $ measures 62 degrees. According to the inscribed angle theorem: $$ \angle ABC = \frac{1}{2} \times \text{measure of arc } BC $$ $$ \angle ABC = \frac{1}{2} \times 62^\circ = 31^\circ $$ Thus, the measure of $ \angle ABC $ is 31 degrees. The correct response is: - $ \angle ABC $ is 31 degrees. $ \triangle ABC $ is 31 degrees.