Orienting Yourself: The Use of Coordinates: questions with answers

The origin lies ON both axes, so it is zero units from each of them. Its x-coordinate and its y-coordinate are therefore both 0, and O is the point $(0, 0)$. It is a point of the plane like any other, not merely the mark where counting begins.

One number is enough on a line, but a plane is two-dimensional: a single distance from the origin is satisfied by a whole ring of points, and a single distance from one axis is satisfied by a whole line of them. Two perpendicular lines are needed, and the point is fixed by its distance from each.

The first coordinate is zero, so P lies on the y-axis. Its distance from O is 8 units. The sign of -8 places P below O.

P has not moved off the y-axis at all, so its distance from the y-axis is 0 and the x-coordinate is 0. It is 7 units below the x-axis, so the y-coordinate is -7. The horizontal reading is written first: $(0, -7)$.

The x-axis is a horizontal mirror, so the image is directly below or above P and its sideways position never changes: the x-coordinate stays -7. The height flips — P was 7 units from the mirror and its image is 7 units from it on the other side — so the y-coordinate turns from -7 into 7. Again it is the mirror's direction that decides, not its name.

The y-axis is a vertical mirror, so the image is the same height as P — it has not risen or fallen at all, and the y-coordinate is untouched at -9. What changes is how far across P is: it was 2 units from the mirror on one side and the image is 2 units from it on the other, so the x-coordinate turns from -2 into 2. The mirror's NAME is not the coordinate that changes; the mirror's DIRECTION is what matters.

Knowing a y-coordinate of is 3 fixes one thing only: which side of the x-axis P lies on. The two quadrants I or II sit on opposite sides of the y-axis, and the side is decided by the SIGN of the x-coordinate. A size says how far P has gone, never which way, so neither the size of the x-coordinate nor the distance from O can choose between them.

Numbering starts in the top right, where both coordinates are positive, and runs ANTICLOCKWISE: I is $(+, +)$, II is $(-, +)$, III is $(-, -)$ and IV is $(+, -)$. Q is negative then positive, which is the pattern of quadrant II.

Read across to the y-axis first: P stands 3 units from it, on the right, so the x-coordinate is 3. Then read down to the x-axis: P stands 2 units from it, below it, so the y-coordinate is -2. The horizontal reading is written first, giving $(3, -2)$. Dropping the minus signs to give $(3, 2)$ keeps the two distances but throws away which side of each axis P is on, which is the only thing the signs were carrying.

Distance from the x-axis is measured by the y-coordinate, with its sign set aside. P's y-coordinate is -2, so P is 2 units from the x-axis — the minus sign says P is BELOW the axis, not that the distance is negative. The x-coordinate 4 measures the distance from the OTHER axis and has no part in this answer.

A and B have the same y-coordinate, 8, so AB is horizontal and its length is the gap between the x-coordinates: $10 - 1 = 9$. Subtracting the y-coordinates instead gives 0, which is the vertical rise of a horizontal segment, not its length.

Both points have y-coordinate 4, so AB is horizontal. Here A is named first but lies to the RIGHT of B. Taking B's x-coordinate minus A's gives $1 - 9 = -8$, and a length cannot be negative. A length is the larger x-coordinate minus the smaller: $9 - 1 = 8$ units.

The shifts are 1 and 4, so the distance squared is $1^2 + 4^2 = 17$ and the distance is $\sqrt{17}$. 17 is not a perfect square, so this surd IS the answer — it cannot be tidied into a whole number, and writing 17 instead would be giving the square of the distance. Squaring the coordinates before subtracting them gives 63 under the root, which is a different number entirely: the formula subtracts FIRST and squares afterwards.

The horizontal and vertical distances from O are 5 and 12 units. By Pythagoras, the distance is $\sqrt{169} = 13$ units. Adding those distances gives 17, the length of a route along the axes. Subtracting their squares before taking a square root gives $\sqrt{119}$; the squares must be added.

The distance from O comes from squaring both coordinates and adding them. Swapping only changes the order of that addition, so Q is the same distance from O as P. P has one positive and one negative coordinate. Swapping moves the minus sign to the other coordinate, so Q lies in quadrant II, not quadrant IV.