Question

what type of segment is \\(\overline{dh}\\)?

• radius
• diameter
• chord
• center

Explanation:

Identify the circles and points

The diagram shows a larger circle with points \(C, D, E, F, G, B\) on its circumference, and a smaller circle passing through points \(A, B, H, F\). Point \(H\) is the center of the smaller circle, and point \(G\) is on the larger circle's boundary. Segment \(\overline{DG}\) is a chord of the larger circle that passes through the center \(H\) of the smaller circle.

Analyze segment DH

Point \(H\) is the center of the smaller circle, and point \(D\) lies on the boundary of the larger circle. For the smaller circle, \(H\) is the center, and \(D\) is an external point. For the larger circle, \(D\) and \(G\) are on the boundary, making \(\overline{DG}\) a chord, while \(H\) is an interior point (specifically the center of the smaller intersecting circle). Point \(H\) is the center of the smaller circle, and point \(D\) is on its boundary. Therefore, the segment \(\overline{DH}\) connects the center of the smaller circle \(H\) to a point \(D\) on the larger circle, but looking closely at the smaller circle, its boundary passes through \(H\) as an intersection? No, the arc of the smaller circle has center \(A\). Let's re-examine: the circle on the right has center \(A\), and its boundary passes through \(H\), \(B\), \(F\). Thus, \(A\) is the center, and \(\overline{AH}\) is a radius.
Let's look at the segment \(\overline{DH}\). Point \(H\) is the center of the larger circle? No, the line \(\overline{DG}\) passes through \(H\). If \(H\) is the center of the larger circle, then \(\overline{DH}\) connects the center \(H\) to a point \(D\) on the circle, which makes \(\overline{DH}\) a radius of the larger circle. Since \(\overline{DG}\) is a straight line segment passing through \(H\) from one side of the circle to the other, \(\overline{DG}\) is a diameter, and its half \(\overline{DH}\) is a radius.

Match with options

Since \(H\) is the center of the larger circle, the segment \(\overline{DH}\) connecting the center \(H\) to the point \(D\) on the circle is a radius.

Answer:

• (A) Radius (Correct answer)
• (B) Diameter
• (C) Chord
• (D) Center