Vertices

Definition

The vertices of a geometric shape refer to the points where two or more line segments, curves, or edges meet. They are the corners or points that define the shape and its boundaries.

5 Must Know Facts For Your Next Test

1. The vertices of an ellipse are the points on the ellipse where the curve changes direction and are the points farthest from the center of the ellipse.
2. The vertices of a hyperbola are the points where the branches of the hyperbola intersect the major axis, and they are the points farthest from the center of the hyperbola.
3. The distance between the vertices of an ellipse is the major axis, while the distance between the vertices of a hyperbola is the transverse axis.
4. The coordinates of the vertices of an ellipse are determined by the length of the major and minor axes, while the coordinates of the vertices of a hyperbola are determined by the length of the transverse and conjugate axes.
5. The vertices of an ellipse or hyperbola play a crucial role in determining the shape, size, and other properties of these conic sections.

Review Questions

• Explain the relationship between the vertices and the major and minor axes of an ellipse.
  • The vertices of an ellipse are the points on the ellipse where the curve changes direction and are the points farthest from the center of the ellipse. The distance between the vertices of an ellipse is the length of the major axis, while the distance between the vertices on the minor axis is the length of the minor axis. The coordinates of the vertices are determined by the lengths of the major and minor axes, and the vertices are essential in defining the shape and size of the ellipse.
• Describe the role of the vertices in determining the properties of a hyperbola.
  • The vertices of a hyperbola are the points where the branches of the hyperbola intersect the major axis, and they are the points farthest from the center of the hyperbola. The distance between the vertices of a hyperbola is the length of the transverse axis, while the distance between the vertices on the conjugate axis is the length of the conjugate axis. The coordinates of the vertices are determined by the lengths of the transverse and conjugate axes, and the vertices are crucial in defining the shape, size, and other properties of the hyperbola, such as the location of the foci and the asymptotes.
• Analyze how the vertices of an ellipse or hyperbola relate to the eccentricity and shape of the conic section.
  • The vertices of an ellipse or hyperbola are directly related to the eccentricity and overall shape of the conic section. The distance between the vertices, which corresponds to the major or transverse axis, is a key factor in determining the eccentricity. A larger distance between the vertices results in a more elongated, or eccentric, shape, while a smaller distance between the vertices leads to a more circular or flattened shape. Additionally, the location of the vertices relative to the center of the ellipse or hyperbola influences the curvature and overall appearance of the conic section. By understanding the relationship between the vertices and the eccentricity, one can better visualize and analyze the properties of these important geometric shapes.

Related terms

Foci:

The foci of an ellipse or hyperbola are the two special points within the shape that define its eccentricity and other properties.

Eccentricity:

The eccentricity of an ellipse or hyperbola is a measure of how elongated or flattened the shape is, and is related to the location of the foci.

Asymptotes:

The asymptotes of a hyperbola are the two straight lines that the branches of the hyperbola approach but never touch, and are determined by the location of the vertices and foci.