Point A(1, 6) is □ units from the y-axis and □ units from the x-axis. Fill in each blank

1, 6. The distance from the y-axis runs across, so it comes from x = 1: 1 unit. From the x-axis it runs up or down, from y = 6: 6 units. A distance is never negative.

(−1, 1) turned 180° about the origin goes to (□, □). Fill in each blank

1, −1. A 180° turn takes it to the other side of the origin: both signs change, (1, −1).

Coordinates and trigonometry · Ages 12–13

First steps in coordinates

From “Reading and plotting points” to “Rotation about the origin”, step by step, each skill with its methods, worked examples and answers.

Skills and methods in this topic, easy to hard

Reading and plotting pointsAcross first, then up or down: (x, y)
Method(−3, 2): 3 left, 2 up
Another way: Look to the axesDown to the x-axis for x, across to the y-axis for y
Quadrants and points on the axesThe signs give the quadrant; a 0 puts it on an axis
Method(−4, 5): (−, +), the second quadrant
Another way: Plot and look(−4, 5) lies up and left of the origin
Distance along a horizontal or vertical lineLarger minus smaller: 3 − (−4) = 7
Method(−4, 2) and (3, 2): 3 − (−4) = 7
Another way: Add the two parts4 + 3 = 7
Areas on the coordinate planeSplit and add, or a rectangle round it minus the corners
MethodBase 5, height 5: 12 × 5 × 5 = 12.5
Another way: Rectangle minus corners6 × 5 − 9 − 4 − 5 = 12
TranslationLeft and right change x; up and down change y
Method(−2, 3), 5 right and 4 down: (3, −1)
Another way: Count squaresEvery corner moves the same squares
ReflectionIn the x-axis x stays; in the y-axis y stays
Method(3, −2): in the x-axis (3, 2), in the y-axis (−3, −2)
Another way: Count to the mirror(1, 2) in x = 3: 2 to the line, 2 beyond, (5, 2)
Rotation about the origin180°: both signs change; 90°: swap, then change one sign
Method(4, 1): 90° anticlockwise (−1, 4); clockwise (1, −4)
Another way: Turn the right triangle4 across, 1 up turns to 4 up, 1 left

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

Example 1Reading and plotting points →
Point A(1, 6) is units from the y-axis and units from the x-axis.
Fill in each blank
Show the answer and working
Answer: 1, 6
The distance from the y-axis runs across, so it comes from x = 1: 1 unit. From the x-axis it runs up or down, from y = 6: 6 units. A distance is never negative.
Example 2Areas on the coordinate plane →
Trapezium ABCD has corners A(−2, −4), B(6, −4), C(6, 3) and D(−4, 3). Its area is square units.
Show the answer and working
Answer: 63
AB and DC are both horizontal, the parallel sides: AB = 8, DC = 10, height 7; area = 12 × (8 + 10) × 7 = 63.
Example 3Rotation about the origin →
(−1, 1) turned 180° about the origin goes to (, ).
Fill in each blank
Show the answer and working
Answer: 1, −1
A 180° turn takes it to the other side of the origin: both signs change, (1, −1).

Free worksheet: First steps in coordinatesWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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