Geometry Distance and Midpoint Formulas

Distance Formula

• The distance between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane is given by the formula: √((x₂ - x₁)² + (y₂ - y₁)²).
• This formula is a direct application of the Pythagorean theorem.
• The distance is always a non-negative value.
• Example: Find the distance between points A(2, 3) and B(5, 7).
  • distance = √((5 - 2)² + (7 - 3)²) = √(3² + 4²) = √(9 + 16) = √25 = 5

Midpoint Formula

• The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by the formula: ((x₁ + x₂)/2, (y₁ + y₂)/2).
• This formula finds the point exactly halfway between the two endpoints.
• Example: Find the midpoint of the line segment with endpoints (-3, 1) and (7, 9).
  • midpoint = ((-3 + 7)/2, (1 + 9)/2) = (2, 5)

Equation of a Circle

• A circle is the set of all points in a plane that are equidistant from a fixed point called the center.
• The distance from the center to any point on the circle is the radius (r).
• The standard equation of a circle with center (h, k) and radius r is: (x - h)² + (y - k)² = r²
• Example: The equation (x - 2)² + (y + 3)² = 16 represents a circle with center (2, -3) and radius 4.

Placing Figures in a Coordinate Plane

• Choosing suitable coordinate axes and origin is crucial.
• Figures can be positioned anywhere in the coordinate system.
• To accurately position a figure, key points or vertices of the shape need specific coordinates.
• Example
  • Place a rectangle with vertices at (-1, 2), (4, 2), (4, -2), (-1, -2) in a coordinate plane. This visually places the rectangle.
• Consider the scale of the coordinate plane and the size of the figure, this helps to determine the positioning efficiency.
• When placing shapes, using coordinate geometry formulas (perpendicular bisectors, etc.) may help in positioning figures with certain conditions.
• When dealing with multiple figures, consider arranging them strategically to enhance clarity and understanding within the coordinate plane.

This quiz covers the distance and midpoint formulas used in coordinate geometry. It includes examples and applications to reinforce understanding of these essential concepts. Test your knowledge on the principles of circles and their equations as well.