Coordinates and Line Geometry
Coordinates describe the exact position of a point on a grid using an ordered pair (x, y), where the x-coordinate gives the horizontal position and the y-coordinate gives the vertical position, measured from the origin (0, 0), with both axes extending into negative numbers to split a full grid into four quadrants. Coordinates also describe straight line segments: the midpoint is the point exactly halfway between two coordinates, and the gradient, the change in y divided by the change in x, measures how steep the segment is.
Method
- Read or plot a coordinate as (x, y): move x units horizontally from the origin (right if positive, left if negative), then y units vertically (up if positive, down if negative).
- To find the midpoint of a line segment joining (x1, y1) and (x2, y2), average the two x-coordinates and average the two y-coordinates: midpoint = ((x1+x2)/2, (y1+y2)/2).
- To find the gradient of a line segment joining (x1, y1) and (x2, y2), divide the change in y by the change in x: gradient = (y2-y1)/(x2-x1).
- A positive gradient slopes upward from left to right; a negative gradient slopes downward from left to right; a horizontal line has a gradient of 0.
- For a horizontal segment, the distance between the two points is simply the difference between their x-coordinates (ignoring the sign); for a vertical segment, it is the difference between their y-coordinates.
- Always subtract the coordinates in the same order in both the numerator and the denominator of the gradient calculation, to avoid a sign error.
Worked example
A line segment joins point A(-4, 1) to point B(6, 7). Find (a) the midpoint of AB, (b) the gradient of AB.
- Midpoint x-coordinate: (-4 + 6)/2 = 2/2 = 1.
- Midpoint y-coordinate: (1 + 7)/2 = 8/2 = 4. So the midpoint of AB is (1, 4).
- Gradient: change in y = 7 - 1 = 6; change in x = 6 - (-4) = 10.
- Gradient = 6/10 = 3/5, so the gradient of AB is 3/5 (or 0.6).
Practice questions
Try each question, then tap to reveal the answer.
Q1Write down the coordinates of the point that is 5 units to the right and 3 units down from the origin.Show answer
Answer: (5, -3)
Q2Find the midpoint of the line segment joining (2, 8) and (10, 2).Show answer
Answer: ((2+10)/2, (8+2)/2) = (6, 5)
Q3Find the midpoint of the line segment joining (-6, 3) and (4, -9).Show answer
Answer: ((-6+4)/2, (3+(-9))/2) = (-1, -3)
Q4Find the gradient of the line segment joining (1, 2) and (5, 14).Show answer
Answer: (14-2)/(5-1) = 12/4 = 3
Q5Find the gradient of the line segment joining (-2, 9) and (3, -1).Show answer
Answer: (-1-9)/(3-(-2)) = -10/5 = -2
Q6Point P is at (3, 7) and point Q is at (3, -5). Find the distance PQ.Show answer
Answer: Both points share x = 3, so PQ is vertical; distance = 7 - (-5) = 12 units
Q7M(2, 1) is the midpoint of the line segment joining A(-3, -3) and B. Find the coordinates of B.Show answer
Answer: B_x = 2x2 - (-3) = 7 and B_y = 2x1 - (-3) = 5, so B = (7, 5)
Q8A line segment has gradient 2 and joins (0, 1) to (4, k). Find the value of k.Show answer
Answer: (k-1)/(4-0) = 2, so k - 1 = 8, and k = 9
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
Point A has coordinates (-5, 2) and point B has coordinates (3, -4). (a) Find the midpoint of the line segment AB. (b) Find the gradient of the line segment AB.
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A quadrilateral has vertices A(1, 1), B(7, 1), C(7, 5) and D(1, 5). (a) Find the coordinates of the midpoint of the diagonal AC. (b) Find the gradient of the side AB. (c) Explain why the gradient of AB is 0.
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Free printable worksheet
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This topic is chapter 13 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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