Unit 11: Coordinate Geometry
Points, lines, and circles rewritten as equations, and how a proof can be run entirely with algebra instead of a compass.
7 topics · 23 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
The Distance Formula
- How it worksHow does the distance formula come from applying the Pythagorean theorem to a horizontal and vertical leg?
- Worked exampleFind the distance between the points (1, 2) and (4, 6).
- Common mistakeWhy isn't the distance between two points just the difference of their x-coordinates plus the difference of their y-coordinates?
- Going furtherWhat happens to the distance formula when both points share the same y-coordinate?
Your turn1 of 2
Find the distance between the points (4, 3) and (9, -4).The Midpoint Formula
- How it worksHow does averaging the x-coordinates and the y-coordinates separately locate the midpoint of any segment?
- Worked exampleFind the midpoint of the segment connecting (-3, 5) and (7, 1).
- Going furtherWhat happens to the midpoint formula when it's used to find the center of a circle from two endpoints of a diameter?
Your turn1 of 2
Find the sum of the coordinates of the midpoint of the segment connecting (4, 3) and (9, -4).Slope, Parallel Lines, and Perpendicular Lines
- How it worksHow does multiplying two perpendicular lines' slopes always give negative one?
- Worked exampleShow whether the lines through (0, 0), (2, 4) and (1, 5), (3, 9) are parallel, perpendicular, or neither.
- Common mistakeWhy doesn't a vertical line have a slope of zero?
Your turn1 of 2
A line passes through (0, 0) and (11, 4). Find its slope.Distance from a Point to a Line
- How it worksHow does the perpendicular segment from a point to a line become the shortest of every possible segment you could draw?
- Worked exampleFind the distance from the point (3, 4) to the line y = 1.
- Ties togetherHow does the distance from a point to a line reuse the same perpendicular idea behind an altitude of a triangle?
Your turn1 of 2
Find the distance from the point (6, 6) to the horizontal line y = 17.Coordinate Proofs
- How it worksHow does using variables for coordinates instead of numbers let one proof cover every possible triangle of that type?
- Worked exampleProve using coordinates that the diagonals of a rectangle with vertices (0,0), (a,0), (a,b), (0,b) are congruent.
- Common mistakeWhy doesn't picking one specific numerical example ever count as a full coordinate proof?
Your turn1 of 2
A rectangle has vertices (0, 0), (15, 0), (15, 14), and (0, 14). Find the length of each diagonal (they are congruent, so either one works).Partitioning a Segment in a Given Ratio
- How it worksHow does a weighted average of the endpoint coordinates locate a point that divides a segment in a given ratio?
- Worked exampleFind the point that divides the segment from (0, 0) to (10, 5) in a 3 to 2 ratio.
- Ties togetherHow does the midpoint formula turn out to be just this ratio formula's special case at exactly 1 to 1?
Your turn1 of 2
Find the sum of the coordinates of the point that divides the segment from (0, 0) to (22, 21) in a 6 to 2 ratio, measured from the first point.The Equation of a Circle
- How it worksHow does the distance formula from a center point to any point on the circle turn directly into the circle's equation?
- Worked exampleFind the equation of a circle with center (2, -3) and radius 5.
- Worked exampleFind the center and radius of a circle from the equation x2 + y2 - 6x + 4y - 12 = 0.
- Common mistakeWhy doesn't the equation of a circle look like a linear equation the way a line's does?
Your turn1 of 2
A circle has center (4, 3) and radius 14. Find the value of (x - h) squared plus (y - k) squared for the point (x, y) = (-7, -13).