2.3.1 · D5Coordinate Geometry
Question bank — Cartesian plane — axes, quadrants, ordered pairs
Reminder of the vocabulary you will lean on here, all built in the parent note:
- axis — one of the two perpendicular number lines.
- origin — the crossing point , where both measurements read zero.
- ordered pair — first number = left/right, second = up/down; the order is sacred.
- quadrant — one of the four regions cut out by the axes, where both coordinates are non-zero.
True or false — justify
The point lies in Quadrant I because .
False. Its -coordinate is , so it sits on the -axis, which is a boundary belonging to no quadrant — a quadrant needs both coordinates non-zero.
and are the same point since they use the same two numbers.
False. The pair is ordered: means 3 right then 5 up, means 5 right then 3 up — two genuinely different locations.
Every point of the form lies on a straight line through the origin.
True. Whenever the right/up amounts are equal you get the diagonal line , which passes through at .
The origin belongs to all four quadrants at once.
False. The origin has both coordinates zero, so it fails the "non-zero" test for every quadrant — it belongs to none.
If a point has a negative and a negative , it must be in Quadrant III.
True. Left () plus down () is exactly the bottom-left region, which is Quadrant III by the counterclockwise numbering.
Moving from Quadrant II to Quadrant III, only the sign of changes.
False. Quadrant II is and III is : stays negative, it is the -sign that flips from positive to negative.
The point and the point are reflections of each other across the origin.
True. Negating both coordinates flips a point through the origin (a turn), so these two are mirror images about .
A point can lie on the -axis and the -axis at the same time.
True, but only one such point exists: the origin , where both zeros are satisfied simultaneously.
Spot the error
" is 3 units up and 2 units right, so ."
Error: the horizontal amount always comes first. Right = 2 is the , up = 3 is the , so .
" is in Quadrant II because it is above the -axis and near the left."
Error: means the point sits exactly on the -axis, a boundary — it is in no quadrant regardless of how it looks.
"To plot I go 3 right and 4 up."
Error: negative means the opposite direction. is 3 units left and is 4 units down.
"Quadrants go clockwise: top-right, top-left is II, then bottom-right is III."
Error: numbering is counterclockwise. After top-right (I) and top-left (II) comes bottom-left (III), then bottom-right (IV).
"Distance from to is plus ."
Error: you cannot just add the horizontal and vertical gaps. They are the two legs of a right triangle, so you combine them with Pythagoras: . See Distance formula.
"The point is in Quadrant II because it has a minus sign."
Error: check both signs. (right) and (down) puts it in Quadrant IV, not II.
"A point on the negative -axis, like , is in Quadrant III since is negative."
Error: makes it a boundary point on the -axis, so it is in no quadrant — the region label never applies once a coordinate is zero.
Why questions
Why does order matter in an ordered pair when a set like does not care about order?
Because a coordinate carries a role: the first slot always means horizontal, the second always means vertical. Swapping them re-aims the instructions and lands you somewhere else.
Why are the quadrants numbered counterclockwise instead of like reading a page?
To match the direction of positive rotation used everywhere later — angles and the unit circle also sweep counterclockwise, so the numbering stays consistent with Polar coordinates.
Why do points on the axes belong to no quadrant?
A quadrant is defined by a definite sign of both coordinates. On an axis one coordinate is , which is neither positive nor negative, so no sign pattern fits.
Why can two perpendicular number lines pin down every point in the plane?
Because a plane is two-dimensional: you need exactly two independent pieces of information (how far across, how far up), and the two axes supply precisely those two.
Why do we need the absolute value when reading the horizontal leg for distance?
A length can never be negative. The subtraction might come out negative depending on which point you call first, so the absolute value keeps the leg a genuine positive length.
Why does negating both coordinates rotate a point about the origin rather than reflect it across an axis?
Flipping the -sign alone reflects across the -axis, and flipping alone reflects across the -axis; doing both stacks those two reflections, and two perpendicular reflections combine into a half-turn.
Edge cases
Where is a point whose -coordinate is but -coordinate is any non-zero number?
It lies on the -axis — above the origin if , below it if — and in no quadrant.
Where does live as runs from very negative to very positive?
It slides along the whole -axis: far left for large negative , through the origin at , to far right for large positive — always a boundary point.
What is the "quadrant" of the origin, and what makes it special?
The origin has both coordinates zero, so it is the single point sitting on both axes at once — the shared boundary of all four quadrants and member of none.
As a point moves so that shrinks toward while stays positive, what happens to its quadrant?
It stays in a quadrant (I or II depending on 's sign) right up until the instant , at which moment it lands on the -axis and momentarily belongs to no quadrant.
Can a single point be counted in two quadrants if it is "close" to the axis?
No. Nearness never matters — only the exact sign of each coordinate does. A point is in exactly one quadrant, or on a boundary, never in two.
What quadrant contains points where and are equal and negative, e.g. ?
Quadrant III, because both are negative (left and down) — the diagonal passes through Quadrants I and III only.
Recall Fast self-test
Is in a quadrant? ::: No — it is the origin, on both axes, belonging to no quadrant. Which quadrant is ? ::: Quadrant II (top-left). Does swapping to change its quadrant? ::: Yes — IV becomes II, because order and signs both changed.