Coordinates, Reflection and Translation
Coordinates, reflection and translation are three related topics in 11+ geometry: a coordinate describes the exact position of a point on a grid as an ordered pair, (x, y), measured from the origin (0, 0), where the x-coordinate gives the horizontal position and the y-coordinate gives the vertical position. 11+ papers test plotting and reading coordinates in all four quadrants (including negative values), reflecting a shape in a mirror line such as the x-axis, the y-axis or a line like x = 3, and translating a shape by a given vector.
Before you start
No specific prerequisites - this is a good place to start.
Method
- Remember a coordinate is always written (x, y): the x-coordinate, the horizontal position, comes first, then the y-coordinate, the vertical position.
- In a grid with negative values, moving right or up from the origin is positive, and moving left or down is negative.
- To reflect a point in the x-axis, keep the x-coordinate the same and change the sign of the y-coordinate; to reflect in the y-axis, keep the y-coordinate the same and change the sign of the x-coordinate.
- To reflect in a vertical or horizontal line such as x = 3 or y = -1, find the point's distance from that line and place the image the same distance on the other side.
- To translate a shape, add the horizontal and vertical parts of the given vector to every coordinate of the original shape.
- Always give coordinates as an ordered pair in brackets, and check a reflected or translated point is the correct distance from the mirror line, or the original point, before finalising your answer.
Worked example
Point A is at (2, 5). (a) Write down the coordinates of A after it is reflected in the y-axis. (b) The reflected point is then translated 3 right and 4 down. Write down the coordinates after this translation.
- To reflect in the y-axis, keep the y-coordinate the same and change the sign of the x-coordinate: (2, 5) becomes (-2, 5). This is the answer to (a).
- To translate, add the horizontal part of the movement to the x-coordinate, and the vertical part to the y-coordinate.
- Starting from (-2, 5): the x-coordinate becomes -2 + 3 = 1, and the y-coordinate becomes 5 - 4 = 1.
- The new coordinates are (1, 1). This is the answer to (b).
- Check: the reflection kept the same distance from the y-axis, 2 units, but on the opposite side, and the translation moved exactly 3 right and 4 down, confirming (1, 1) is correct.
Practice questions
Try each question, then tap to reveal the answer.
Q1Write down the coordinates of the point 4 units to the right and 3 units up from the origin.Show answer
Answer: (4, 3)
Q2Point B is at (-3, 2). Reflect B in the x-axis. Write down the new coordinates.Show answer
Answer: (-3, -2)
Q3Point C is at (5, -1). Reflect C in the y-axis. Write down the new coordinates.Show answer
Answer: (-5, -1)
Q4Point D is at (1, 4). Translate D by 2 right and 3 down. Write down the new coordinates.Show answer
Answer: (3, 1)
Q5Point E is at (-2, -5). Translate E by 4 right and 6 up. Write down the new coordinates.Show answer
Answer: (2, 1)
Q6Point F is at (2, 2). Reflect F in the line x = 6. Write down the new coordinates.Show answer
Answer: (10, 2) (F is 4 units to the left of the line, so its image is 4 units to the right of the line)
Q7A point G is reflected in the x-axis to give the image (7, -3). What were the original coordinates of G?Show answer
Answer: (7, 3) (reflecting in the x-axis flips the sign of the y-coordinate, so reversing it flips the sign back)
Q8Point H is at the origin, (0, 0). Translate H by 3 left and 5 up. Write down the new coordinates.Show answer
Answer: (-3, 5)
Exam-style questions
Written in the style of a 11+ exam paper, with a full mark scheme.
Triangle T has vertices at A(1, 1), B(4, 1) and C(1, 3). Triangle T is reflected in the y-axis to give triangle T'. Write down the coordinates of the image of each vertex, A', B' and C'.
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Point P is at (-4, 3). (a) P is reflected in the x-axis to give point Q. Write down the coordinates of Q. (b) Q is then translated 5 right and 2 down to give point R. Write down the coordinates of R. (c) State the single vector that would translate P directly to R.
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Free printable worksheet
Want more practice on paper? Download the coordinates, reflection and translation worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 34 of 11+ Maths Workbook 2, the whole course as one free printable PDF.
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