Mathematics Year 6: Turning a point and a shape around a centre
Mathematics, Year 6 (Portugal): Turning a point, a triangle or a quadrilateral about a centre through a given angle and direction, using ruler, compass and protractor.
A centre, an angle and a direction
To turn a point P you need three things: the centre O, the angle and the direction (clockwise or anticlockwise). If P is 4 cm from O and you turn it by 40°, 80° and 120°, you get P₁, P₂ and P₃, all 4 cm from O. So they all lie on a circle with centre O and radius 4 cm: the point only travels round the centre, it never gets closer or further away.
Why the compass belongs in the job
A rotation pins a shape to its centre, like a door pinned to its hinges. Every point moves along a circle around the centre, and its distance from the centre never changes. That is why the compass helps: with its point on O and its opening equal to OP, it draws every place where the image of P could be. The angle tells you which of those places is the right one.
A case: turning triangle ABC by 60° about O
Triangle ABC has OA = 3 cm, OB = 5 cm and OC = 4 cm; measured from ray OA, B is at 20° and C at 65°. Turn it 60° anticlockwise. For each vertex: compass on O, opened to the distance to that vertex; protractor on O, 60° from that vertex; mark the image. A', B' and C' end up at 60°, 80° and 125° from OA. Join them. Check: 80° − 60° = 20° and 125° − 60° = 65°, the same as before. A quadrilateral works the same way, with four vertices.
The angle is measured at O, not at P
The reasonable mistake: putting the protractor on P, because P is the point that will move. But the angle of rotation is the angle POP', with its vertex at the centre O. Another mistake: marking P' in the right direction but at a different distance from O; then the point has left the circle and the shape gets distorted. Rule: protractor and compass on O, and at the end check that OP' = OP.
Where you see it: clock hands, wheels and geometry software
The minute hand of a clock is a rotation happening: from 12 to 3 it turns 90° (360° ÷ 4), from 12 to 4 it turns 120°, always clockwise, and its tip draws a circle. The same is true of a point on the rim of a bicycle wheel. In dynamic geometry software you choose the shape, the centre and the angle, and the program does the construction you did by hand.
Test what you learned
Point P is 5 cm from the centre O. You turn P about O through many different angles. Where do all the images of P lie?
- a) On the straight line through O and P
- b) On a circle with centre P and radius 5 cm
- c) On a circle with centre O and radius 5 cm
- d) On circles of different sizes, bigger for bigger angles
Correct answer: c) On a circle with centre O and radius 5 cm — Yes: turning never changes the distance to O, so every image is 5 cm from O. That describes a circle centred at O.
OP = 4 cm. Turning P about O by 40° gives P₁; turning it by 130° in the same direction gives P₂. What is angle P₁OP₂, and how far is P₂ from O?
- a) 170° and 4 cm
- b) 90° and 4 cm
- c) 130° and 4 cm
- d) 40° and 4 cm
Correct answer: b) 90° and 4 cm — Right: 130° − 40° = 90°, and a rotation never changes the distance to O, so P₂ is still 4 cm from O.
You must turn point P by 60° about O using a protractor and a compass. Where do they go?
- a) Both on O; the compass, opened to OP, keeps P' at the same distance
- b) Both on P, measuring 60° from PO
- c) Protractor on O only; the angle alone places P'
- d) Protractor on the midpoint of PO, compass on P
Correct answer: a) Both on O; the compass, opened to OP, keeps P' at the same distance — Right: the angle is measured at the centre O, and the compass on O keeps OP' = OP, so P' stays on the circle.
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