Question

ANALYZING RELATIONSHIPS Suppose the graph of a line that passes through the origin intersects the graph of a circle with its center at the origin. When you know one of the points of intersection, explain how you can find the other point of intersection without performing any calculations.

Short answer

The other point of intersection between the line and the circle would be the mirror image of the given intersection point across the origin, which would be (-x, -y) if the given intersection point is (x, y).

Step-by-step solution

Understanding the intersection point

If a line and a circle intersect at a point, this point would have the coordinate (x, y). This is the first intersection point given.

Understanding the properties of a circle

A circle is a set of points in a plane that are a certain distance from a certain point called the center. In geometric terms, a circle is perfectly symmetrical, and this property will also be reflected on a coordinate plane.

Use the symmetry of the circle

Because of this symmetry, and given that our line passes through the origin, we can conclude that the other intersection point is the mirror image point across the origin. That is, if one intersection point is at (x, y), the other intersection point will be at (-x, -y).

Key concepts

Intersection Points

The concept of intersection points is central in many areas of geometry, especially when studying the relationships between different shapes like lines and circles. When a line intersects a circle, the point(s) where the line and circle meet are known as intersection points.
These are the coordinates that satisfy the equations of both the circle and the line. In simpler terms, they are the points where the shapes "touch" on a coordinate plane.
For example, suppose you have a line and a circle on a graph, both centered at the origin. If the line crosses the circle, that crossing happens at one or more specific points, and these are the intersection points.
This concept helps us understand the spatial relationship between these geometric figures. Understanding intersection points is key in problems where you analyze or predict the points of contact between a line and other curves like circles.

Circle Symmetry

A defining property of circles is their symmetry. Symmetry means that every point on a circle is evenly balanced around the center. This extensive symmetry implies that any line through the center will equally divide the circle into two identical halves.
Given that the line and circle both pass through the origin, an intersection point at (x, y) will automatically have a symmetrical counterpart at (-x, -y). This occurs because the line acts as an axis of symmetry for the circle.
Understanding circle symmetry can be very useful in solving geometric problems. For instance, when you know one intersection point, you can leverage this symmetry to immediately determine its counterpart without any complex calculations.

Coordinate Geometry

Coordinate geometry, also called analytic geometry, allows us to understand the location and relationships of geometrical figures through numerical coordinates.
In the plane, geometry is about positions and shapes, but coordinate geometry gives it a numerical context by using an (x, y) system.
By plotting points on this coordinate plane, we can graph lines, circles, and many other geometric shapes, facilitating easier analysis and visualization of relationships like intersections and symmetries.

• This allows for the exact determination of intersection points between different figures.
• We can rely on coordinate symmetry properties to predict or calculate unknown values efficiently.
• Practically, it aids in translating intuitive geometric relationships into calculable equations and variables.

Hence, coordinate geometry is a comprehensive tool in geometrical problem-solving.