Question

in the image, point a is the center of the circle. which two line segments must be equal in length?
a. \\(\overline{ah}\\) and \\(\overline{bc}\\)
b. \\(\overline{ef}\\) and \\(\overline{a}\\)
c. \\(\overline{h}\\) and \\(\overline{bc}\\)
d. \\(\overline{ef}\\) and \\(\overline{hi}\\)

Explanation:

Step1: Recall the property of congruent chords in a circle

In a circle, chords that are equidistant from the center are congruent (have equal length). Also, if two chords subtend equal central angles, they are congruent.

Step2: Analyze each option

• Option A: \( \overline{AH} \) is a radius (from center \( A \) to a point \( H \) on the circle), and \( \overline{BC} \) is a chord (not a radius). A radius and a non - radius chord (unless the chord is a diameter) are not equal in length.
• Option B: \( \overline{EF} \) is a chord and \( \overline{AI} \) is a radius. A radius and a non - radius chord (unless the chord is a diameter) are not equal in length.
• Option C: \( \overline{H} \) is not a line segment. This is an incorrect notation.
• Option D: \( \overline{EF} \) and \( \overline{HI} \) are both chords of the circle. Since the central angles subtended by \( \overline{EF} \) and \( \overline{HI} \) at the center \( A \) of the circle are equal (by visual inspection of the symmetry of the circle - like the number of grid squares they span in terms of the circular pattern around the center), by the property of congruent chords in a circle (chords with equal central angles are congruent), \( \overline{EF}\) and \( \overline{HI}\) have equal length.

Answer:

D. \( \overline{EF} \) and \( \overline{HI} \)