In circle geometry, what do we call a line segment whose endpoints both lie on the circle?

Introduction / Context:
Circles have several important associated lines and segments, including radii, diameters, chords, tangents and secants. Each term describes a specific geometric configuration. This question focuses on the definition of a chord in circle geometry.

Given Data / Assumptions:
- We are working with a circle in a plane.
- We consider a line segment whose two endpoints lie on the circumference of the circle.
- We must choose the correct term for such a segment from the given options.

Concept / Approach:
A chord is defined as a line segment with both endpoints on the circle. The diameter is a special type of chord that passes through the center of the circle. By contrast, a secant is a line (not just a segment) that intersects the circle at two points, and a tangent is a line that touches the circle at exactly one point. Recognizing these distinctions allows us to select the proper term.

Step-by-Step Solution:
Step 1: Identify the key condition in the question: both endpoints of the segment lie on the circle. Step 2: Recall that a chord is defined as a line segment joining two points on the circle. Step 3: A diameter is a chord that passes through the center, but the question does not require the center to be involved, so the more general term chord is appropriate. Step 4: A secant is an infinite line that cuts the circle at two points, not just a bounded segment. Step 5: A tangent touches the circle at exactly one point and does not have two endpoints on the circle. Step 6: Therefore, the correct term is chord.

Verification / Alternative Check:
Draw a circle and mark two distinct points on its circumference. Connect these points with a straight line segment. Geometry textbooks uniformly refer to this segment as a chord. You can further draw several such segments to observe that they all match the definition of a chord, whether or not they pass through the center.

Why Other Options Are Wrong:
Option A (secant) refers to the entire line that passes through the circle at two points, extending beyond the circle on both sides. It is not limited to the segment between the intersection points.
Option C (tangent) refers to a line that touches the circle at exactly one point, not two.
Option D (None of the above) is wrong because chord is a valid and correct choice among the options.

Common Pitfalls:
Students sometimes confuse chord and secant, or think of diameter as a separate concept rather than a special type of chord. Remember that chord always involves two endpoints on the circle and is a finite segment, whereas secant is an entire line extending beyond the circle.

Final Answer:
A line segment whose endpoints are both on the circle is called a chord.