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Coordinate Geometry
Distance, Midpoint & Slope
Coordinate geometry connects algebra and geometry by placing points and shapes on a coordinate plane. It lets students describe location with ordered pairs, measure distance, find midpoints, and write equations for lines. This makes geometry more precise and gives a visual way to solve many algebra problems. It is used in graphing, design, engineering, and physics.
On the coordinate plane, each point is written as (x, y), where x shows horizontal position and y shows vertical position. From these coordinates, students can calculate slope to describe steepness, use the distance formula to measure between points, and apply the midpoint formula to find the center of a segment. Coordinate geometry also helps classify shapes, test whether lines are parallel or perpendicular, and analyze symmetry across the axes.
Key Facts
- A point is written as (x, y), where x is the horizontal coordinate and y is the vertical coordinate.
- Distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
- Midpoint formula: M = ((x1 + x2)/2, (y1 + y2)/2)
- Slope formula: m = (y2 - y1)/(x2 - x1)
- Equation of a line in slope intercept form: y = mx + b
- Parallel lines have equal slopes, and perpendicular lines have slopes whose product is -1.
Vocabulary
- Ordered pair
- An ordered pair is a set of two numbers, written (x, y), that gives the location of a point on the coordinate plane.
- Origin
- The origin is the point (0, 0) where the x-axis and y-axis intersect.
- Slope
- Slope is a number that describes the steepness and direction of a line.
- Midpoint
- The midpoint is the point exactly halfway between the endpoints of a line segment.
- Quadrant
- A quadrant is one of the four regions formed by the x-axis and y-axis on the coordinate plane.
Common Mistakes to Avoid
- Switching the coordinates in an ordered pair, because (x, y) and (y, x) usually name different points on the plane. Always move horizontally first for x and vertically second for y.
- Using the slope formula with mismatched subtraction, because subtracting x-values and y-values in different orders gives the wrong sign. Keep the order consistent: y2 - y1 over x2 - x1.
- Forgetting the square root in the distance formula, because squaring differences alone gives the square of the distance, not the actual distance. After adding the squared differences, take the square root.
- Assuming all perpendicular lines have negative reciprocal slopes, because horizontal and vertical lines are also perpendicular but one has slope 0 and the other has undefined slope. Check the graph and line type before applying the rule.
Practice Questions
- 1 Find the distance between A(2, 3) and B(8, 11).
- 2 Find the midpoint of the segment with endpoints C(-4, 6) and D(10, -2). Then find the slope of CD.
- 3 Line l has slope 3/4 and line k has slope -4/3. Explain whether the lines are parallel, perpendicular, or neither, and justify your answer.