Short Answer 
An arc in geometry is a smooth curve connecting two points on a circle, and its length relates directly to the central angle; as the radius increases, both the angle and arc length increase. Selecting the correct graph, such as **Option D**, aids in visualizing this relationship effectively.
Step 1: Understanding Arc Definition
In geometry, an arc is defined as any smooth curve that connects two points along a circle. It is essential to recognize that not just any curve qualifies as an arc, but one that maintains the property of smoothness and connection between two distinct points. Understanding this foundational concept is crucial when discussing the properties and relationships of arcs in a circular context.
Step 2: Relationship Between Arc Length and Central Angle
The relationship between arc length and the central angle in a circle is fundamental. As the radius (r) of the circle increases, both the central angle and the resulting arc length also increase. This implies that a larger radius allows for a longer arc. Key points about this relationship include:
- When the central angle is greater, the arc length is longer.
- The formula for the maximum arc length is based on the circumference, which is given by 2œAr.
- This maximum occurs when the central angle is equal to 360 degrees (or 2œA radians).
Step 3: Identifying the Correct Graph
To visualize the relationship discussed, it is important to identify the correct graph that represents the arc length in relation to the central angle. In this specific scenario, the correct graph is indicated as Option D. Being able to select the appropriate graphical representation helps solidify understanding of how varying the radius affects both the arc length and central angle. With the proper visual tool, one can easily discern patterns and relationships in circular measurements.