Coordinates describe where a point sits on a flat surface such as a page or a screen. Two number lines at right angles, one horizontal and one vertical, are drawn across the surface, and each point is then located by two measurements: how far along the horizontal line it sits, and how far along the vertical line.
P sits 4 units to the right of the vertical line and 2 units above the horizontal one, so its coordinates are (4,2). Q sits 3 units to the left and 5 units below, and both measurements run in the negative direction of their number line, so both carry a minus sign: Q(−3,−5). The horizontal measurement is always written first.
Checkpoint 1
The points A, B and C are plotted on the grid below.
Write down the coordinates of each point.
The two number lines are the axes: the horizontal one is the x-axis and the vertical one is the y-axis. They cross at their zeros, and that crossing point is the origin. The x-axis runs through every point whose y-coordinate is 0, and the y-axis through every point whose x-coordinate is 0.
The coordinates of a point are the pair (x,y). The x-coordinate is the point's displacement from the origin along the x-axis, and the y-coordinate is its displacement along the y-axis. Each displacement is signed: positive to the right or upwards, negative to the left or downwards. A point is named by a letter and its pair, as in P(4,2), and the origin is (0,0).
The order inside the pair is fixed: the x-coordinate first, then the y-coordinate. Swapping the two numbers gives a different point.
When a point sits on an axis, one of its coordinates is 0. Which one tells you which axis: a 0 in the second position means the point lies on the x-axis, and a 0 in the first position means it lies on the y-axis.
The origin lies on both axes, and it is the only point with both coordinates 0. This test works without a picture: (0,−6) is on the y-axis because its first coordinate is 0, whether or not it fits on the grid in front of you, and (7,0) is on the x-axis because its second coordinate is 0.
Checkpoint 2
Four points are plotted on the grid below. Three more are given only by their coordinates: (0,−6), (7,0) and (−5,2).
Sort the points that lie on an axis. Leave the other points where they are.
The coordinates of a point are its two displacements from the origin. The same two numbers describe a move from any point to any other.
Reading the grid, A(−4,3) and B(2,−2). Going from A to B, the x-coordinate changes from −4 to 2, a change of 2−(−4)=6: B is 6 units in the positive x-direction from A. The y-coordinate changes from 3 to −2, a change of −2−3=−5: B is 5 units in the negative y-direction from A. The size of each change is the number of units moved, and its sign is the direction.
From (x1,y1) to (x2,y2), the displacement is x2−x1 in the x-direction and y2−y1 in the y-direction. A positive value is a move to the right or upwards, a negative value a move to the left or downwards.
The displacement runs from the first point to the second. Going the other way, from B to A, reverses both signs.
Checkpoint 3
The points A and B are plotted on the grid.
Complete the sentence.
The two axes cut the rest of the plane into four regions, the quadrants, numbered I to IV anticlockwise from the upper right.
Which quadrant a point is in depends only on the signs of its coordinates.
A point on an axis has a zero coordinate, so it has no sign pattern: points on the axes, including the origin, lie in no quadrant.
Checkpoint 4
Five points are plotted on the grid. Three more are given only by their coordinates: (−40,7), (0.3,−0.2) and (0,−25).
Sort the points by quadrant.
The grid lines on a diagram mean whatever the numbers on the axes say they mean. When every line is one unit, counting squares gives the coordinates directly. Often only some lines carry numbers, and the lines between them stand for a step you have to work out.
Write down the coordinates of R.
Along the x-axis the numbers go up by 10 from one numbered line to the next, with 5 gaps between them. Each unnumbered line is therefore a step of 10÷5=2. Along the y-axis the numbers go up by 20 with 4 gaps, so each step is 20÷4=5.
R is 2 steps to the right of the line marked 10, and 3 steps above the line marked 20:
So R(14,35). Counting the small squares as units would have given (7,7), which is a different point altogether. The numbers on the axes decide the size of every step.
Checkpoint 5
The points P and Q are plotted on the grid below. Only the major gridlines are numbered.
Write down the coordinates of P and Q.
Two points fix a straight path between them.
The segment [AB] is the straight path from the point A to the point B, including both endpoints. The square brackets mark it as a segment: A alone is a point, and [AB] is the whole path between the two.
A segment has no direction. [AB] and [BA] are the same segment, and it makes no difference which endpoint is named first.
Every segment has a point halfway along it.
Going from A(−5,−2) to B(3,4) means moving 6 units up and 8 units across. The point M sits at exactly half of each of those moves: 6÷2=3 up and 8÷2=4 across from A, which puts it at (−1,1). M is the midpoint of [AB]: the point halfway along the segment, as far from A as from B.
Checkpoint 6
Five points are plotted on the grid.
Select every statement that is true.
The midpoint is the average of the two endpoints. Halving each displacement from A is the same as averaging the coordinates: the midpoint's x-coordinate is halfway between the two x-coordinates, and its y-coordinate halfway between the two y-coordinates.
The midpoint of the segment joining (x1,y1) and (x2,y2) is
where x1 and x2 are the two x-coordinates and y1 and y2 the two y-coordinates.
(2,4) and (6,8)
(4,6)
(−3,1) and (5,1)
(1,1)
(0,−2) and (3,4)
(−5,2) and (−1,−4)
Checkpoint 7
The points A(−3,5) and B(6,−4) are the endpoints of the segment [AB]. Find the coordinates of its midpoint.
The distance between A and B is written AB.
It is the length of the segment [AB]. The brackets make the difference: [AB], with brackets, is the segment, a path; AB, without brackets, is its length, a number.
A segment that runs straight across or straight up can be measured by counting grid units. A slanted segment cannot, but the two displacements between its endpoints can, and they determine its length.
From A(1,1) to B(5,4) is 4 across and 3 up. Those two moves and the segment itself form a right triangle: the moves are the two shorter sides, meeting at a right angle, and [AB] is the hypotenuse. Pythagoras' theorem gives its length:
A negative displacement squares to the same value as a positive one, so the direction of travel does not affect the length. Written for any two points, the same calculation is the distance formula.
The distance between (x1,y1) and (x2,y2) is
where x1−x2 is the displacement across and y1−y2 the displacement up, each squared so that its sign does not matter.
The points A and B are plotted on the grid below. Find AB.
Reading the grid, A(−3,2) and B(4,−2). From A to B, the x-coordinate changes by 4−(−3)=7 and the y-coordinate changes by −2−2=−4. Drawn in, these two moves are the shorter sides of a right triangle whose hypotenuse is [AB], so
√65 is the exact distance; 8.06 is its value to 3 significant figures, and either is an acceptable final answer unless the question asks for one form.
When two points share a coordinate, the segment between them is horizontal or vertical and one displacement is zero. The formula then gives the size of the other displacement, which is what counting along the grid gives directly.
Checkpoint 8
Find the exact distance between the two points in the diagram below.
Exam-style problem 1
The following diagram shows four points A, B, C and D.
State which point(s) lie on
the x-axis,
the y-axis.
Write down the quadrant in which D lies.
The distance between C and D is p units greater than the distance between A and B.
Find the value of p.
Exam-style problem 2
The point B lies 6 units in the positive x-direction from the point A. The point C lies p units in the negative y-direction from B, where p>0.
Write down the size of AB^C.
It is given that AC=2√13.
Find the value of p.
It is further given that the midpoint of [AC] has coordinates (2,1).
Find the coordinates of A.
Nice work reaching the end of Coordinates, midpoint, and distance. Here is what this lesson set out to teach - worth reading back with fresh eyes:
0/8 checkpoints checked off