Read the concepts, practice all 160 questions with hints and instant feedback, race the clock in Quiz Mode, or play a game. Click the grid to plot points — click more than once and they'll join up automatically.
Cartesian plane Two perpendicular number lines — a horizontal x-axis and a vertical y-axis — used together to fix the position of any point.
Origin The point (0, 0) where the x-axis and y-axis cross.
Abscissa The x-coordinate of a point — its distance from the y-axis.
Ordinate The y-coordinate of a point — its distance from the x-axis.
Quadrants The axes split the plane into four regions. I: (+,+) · II: (−,+) · III: (−,−) · IV: (+,−).
On an axis A point on the x-axis has ordinate 0. A point on the y-axis has abscissa 0. The origin lies in no quadrant.
Important Formulas
Distance between two points
d = √[(x₂−x₁)² + (y₂−y₁)²]
Midpoint of a segment
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Concept Card
Example
For P(5, 3): abscissa is 5, ordinate is 3. Quadrant I, since both are positive.
On an axis
On the x-axis, points look like (x, 0). On the y-axis, points look like (0, y).
Worked Examples
Q. Point on x-axis equidistant from A(2,−5), B(−2,9)
Let P(x,0). AP²=BP² → x = −7, so P(−7, 0).
Q. Distance between (2,3) and (4,1)
d = √(4+4) = 2√2 units.
Q. Midpoint of A(−2,8), B(−6,−4)
M = (−4, 2).
Q. Are (1,5),(2,3),(−2,−11) collinear?
AB=√5, AC=√265, BC=√212 — not collinear.
Straight Lines & Slope
Slope (m) How steep a line is: m = (y₂−y₁)/(x₂−x₁). Positive slope rises left→right; negative falls; zero is horizontal; a vertical line has no defined slope.
Slope-intercept form A line can be written y = mx + c, where m is the slope and c is the y-intercept.
x- & y-intercepts The x-intercept is where a line crosses the x-axis (y = 0); the y-intercept is where it crosses the y-axis (x = 0).
Parallel lines Two lines are parallel exactly when their slopes are equal: m₁ = m₂.
Perpendicular lines Two lines are perpendicular exactly when their slopes multiply to −1: m₁ · m₂ = −1.
Collinearity by slope Three points are collinear if the slope between any two pairs of them is the same.
Section Formula
Point dividing AB in ratio m : n (from A)
P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))
Slope of a line through two points
m = (y₂−y₁)/(x₂−x₁)
More Worked Examples
Q. Slope through (1,2) and (4,8)
m = (8−2)/(4−1) = 6/3 = 2.
Q. Point dividing A(1,1), B(7,4) in ratio 1:2
P = ((1·7+2·1)/3, (1·4+2·1)/3) = (3, 2).
Q. Is the line through (0,3),(2,7) parallel to y = 2x − 1?
Translations (slides), reflections and symmetry — the grid shows the working, not just the answer.
How to play
Pick a move above — Translate, or a Reflect option.
C is the point you're shown already moved; D is the point you must move yourself.
Work out where D lands, then type its new x and y into the boxes and hit Check.
Stuck? Tap 🔍 Show me on the grid — it draws the guide lines (arrows for a slide, the mirror line for a reflection) so you can see exactly how D became D′.
Tap New Question → whenever you're ready for another.
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C (given)D (find this one)Answer / image
📍 Plot the Point
🎯 Coordinate Guess
Type the coordinates of the plotted point.
🖱️ Drag & Drop
Drag the chip onto the point on the grid.
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📏 Distance Finder
Two points are marked. Work out the distance between them using the distance formula.
📌 Midpoint Finder
Find the midpoint of the segment joining the two marked points.
➗ Bisect & Trisect
Find the point that divides segment AB in the stated ratio (the section formula).
Section formula — P dividing AB in ratio m : n from A: P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))
✖️ Line Crossing
Two straight lines are shown. Solve the pair of linear equations to find where they cross.
📈 Slope & Y-Intercept
A straight line y = mx + c is drawn. Read its slope and y-intercept off the grid.
