Final answer:
Option C. The vertices of a triangle are points on a plane that serve as the intersection of three lines, coordinates in space, or midpoints of three line segments. In 3D space, they require three coordinates and unit vectors along the Cartesian axes to be defined.
Step-by-step explanation:
The three vertices of a triangle can be described using various elements in geometry and coordinate systems. In the context of geometry, the vertices of a triangle are often considered to be:
- Points on a plane
- Intersection of three lines
- Coordinates in space
- Midpoints of three line segments
Specifically, in three-dimensional space, positions are specified by three coordinates on a set of Cartesian axes. The displacement of one point from another is given by the difference in these coordinates: (Ax, Ay, Az) = (x2 − x1, y2 − y1, z2 − z1). Vectors in Three Dimensions are crucial for locating points in space, requiring three unit vectors for the x, y, and z axes respectively: î for the x-axis, ˇ for the y-axis, and ê for the z-axis. These vectors establish a standard right-handed coordinate system when ordered as x-y-z.
When discussing triangles and their vertices specifically, we are referring to a three-sided figure lying on a plane with three angles adding up to 180 degrees, which is a fundamental property of triangles in Euclidean geometry.