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Geometry - Position, direction, and movement

Grade 3Cambridge (IGCSE)

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Coordinates identify a exact position on a grid. We always read the horizontal value (xx-axis) first, then the vertical value (yy-axis). This is often remembered as 'along the corridor and up the stairs'.

A coordinate grid showing point P at (3,2) with lines indicating movement along the x-axis then up the y-axis.
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The four main compass directions are North (N), South (S), East (E), and West (W). Moving clockwise from North, the order is N →\rightarrow E →\rightarrow S →\rightarrow W.

Compass rose showing North, South, East, and West.
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Turns can be measured in right angles. A quarter turn is 90∘90^\circ, a half turn is 180∘180^\circ, and a full turn is 360∘360^\circ. These turns can be clockwise (the way clock hands move) or anti-clockwise.

A circle with a 90 degree sector marked to show a quarter turn.
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Movement on a grid involves translations. A translation moves every point of a shape the same distance in the same direction, such as '2 squares right and 3 squares up'.

📐Formulae

Position = (x, y)

1 quarter turn=90∘ (right angle)1 \text{ quarter turn} = 90^\circ \text{ (right angle)}

1 half turn=180∘ (2 right angles)1 \text{ half turn} = 180^\circ \text{ (2 right angles)}

1 full turn=360∘ (4 right angles)1 \text{ full turn} = 360^\circ \text{ (4 right angles)}

💡Examples

Problem 1:

A treasure map has a grid. A gold coin is located at 3 units to the right and 5 units up from the starting point (0,0). What are the coordinates of the gold coin?

Solution:

(3,5)(3, 5)

Explanation:

To find coordinates, we list the horizontal distance (x) first and the vertical distance (y) second. Since we move 3 units right and 5 units up, the coordinate is (3, 5).

Problem 2:

Robot 'Z' is facing North. It makes a quarter turn clockwise. Which direction is it facing now?

Solution:

East

Explanation:

A clockwise turn follows the direction of a clock's hands. Starting at North, a 90∘90^\circ (quarter) turn to the right leads to East.

Problem 3:

Point A is at (2,2)(2, 2). If you move Point A 3 squares to the right and 1 square down, what are its new coordinates?

Solution:

(5,1)(5, 1)

Explanation:

Moving right increases the x-coordinate: 2+3=52 + 3 = 5. Moving down decreases the y-coordinate: 2−1=12 - 1 = 1. The new position is (5,1)(5, 1).

Problem 4:

Starting at point B(1,1)B(1, 1), a game piece moves 4 units to the right and 3 units up. What are the new coordinates of the game piece?

Grid showing movement from (1,1) to (5,4).

Solution:

New coordinates = (1+4,1+3)=(5,4)(1 + 4, 1 + 3) = (5, 4)

Explanation:

To move 4 units right, add 4 to the xx-coordinate: 1+4=51 + 4 = 5. To move 3 units up, add 3 to the yy-coordinate: 1+3=41 + 3 = 4. The new location is (5,4)(5, 4).

Problem 5:

A toy car is facing West. If it makes a three-quarter turn clockwise, which direction will it be facing?

Diagram showing a 270 degree clockwise turn from West to South.

Solution:

The car will face South.

Explanation:

Starting at West, a quarter turn clockwise leads to North. A second quarter turn (half turn total) leads to East. A third quarter turn (three-quarter turn total) leads to South.