Say you are looking at a wheel of a vehicle. You see a point of the wheel touching the ground. When you look at the wheel again after some time, you again see a point of the wheel touching the ground. Can you tell if the two points are the same point?
Answer: No. A rotating circular wheel looks exactly the same in every position, so there is no way to tell whether the point touching the ground is the same point as before. The circle has complete rotational symmetry.
Step-by-step solution
Idea: Turning a circle about its centre through any angle gives back the same circle. So after any amount of turning, the wheel’s outline looks unchanged, and nothing about the shape tells us which point is at the bottom.
- Every point on the rim of the wheel is at the same distance (the radius) from the centre of the wheel.½ mark
- When the wheel turns about its centre, each rim point moves to another rim point, still at the same distance from the centre. So the turned wheel covers exactly the same circle as before.½ mark
- This is true whatever the angle of turning: 1°, 37° or 200°. We say the circle has complete rotational symmetry.½ mark
- So the two views of the wheel look identical, and from the shape alone we cannot tell whether the same point is touching the ground. (Only a mark on the wheel, such as a valve or a sticker, would let us tell.)½ mark
No. A circular wheel looks the same after being turned through any angle (complete rotational symmetry), so the shape gives no way of telling whether the two points are the same.
Answer to write in the exam
Every point of the rim is at distance r (radius) from the centre.
A turn about the centre through any angle maps the circle onto itself (complete rotational symmetry).
∴ No; the wheel looks the same in both views, so the two points cannot be told apart.
Common mistakes that cost marks
- Answering “yes, because the wheel has turned a full circle”. We do not know how much it has turned; any amount of turning looks the same.
- Thinking a circle has only a few rotational symmetries (like a square, which has only 90°, 180°, 270°, 360°). A circle has them for every angle.
- Using a feature of a real wheel (spokes, valve, tread) as the reason. The question is about the circular shape itself.
How this can come in the exam
MCQ (1 mark)
Through which angles can a circle be turned about its centre so that it looks exactly the same?
- Only 90°
- Only 180°
- Only multiples of 60°
- Any angle
Show answer
(D) Any angle
Every point of the circle moves to another point at the same distance from the centre, whatever the angle. So a circle has complete rotational symmetry.
Try one yourself
A wheel has 6 equally spaced spokes. Through what smallest angle can it be turned so that it looks exactly the same?
Show answer
360° ÷ 6 = 60°. The rim alone would look the same after any turn, but the spokes only line up again after turns of 60°, 120°, 180° and so on.
More questions like this
- Draw a circle on the paper and cut along the circle. Fold the circular paper so that the boundaries overlap, then open it. You see a crease; it is a line of reflection symmetry of the circle. Does this line pass through the centre of the circle?
- 1. What are the rotational symmetries of a square? How many lines of reflection symmetry does it have? What about a regular pentagon? A regular hexagon?
2. What is the length of the longest chord in a circle of radius 5 units? Is there a smallest chord?
3. The locus of points at a given distance from a given point is a circle. What can we say about the locus of points equidistant from two given points?
(Hint: We know that any point that is equidistant from two given points A and B lies on the perpendicular bisector of AB. Does this make the perpendicular bisector the locus? For this, we have to show that all the points on the perpendicular bisector are equidistant from A and B.) - 1. How many circles pass through two points on a plane?
2. Are there circles of all possible radii passing through A and B? What is the radius of the smallest circle passing through A and B? What is the radius of the largest circle passing through A and B?
3. As you move away from segment AB along its perpendicular bisector, do the radii of the circles containing A and B increase or decrease?
4. As you go along the perpendicular bisector, will the circle drawn from that point through A and B appear more curved or less curved?
5. You are given two points A and B on a plane. How many squares can you draw on the same plane with A and B on the boundary? How many squares can you draw on the plane with A and B as the corners of the square? - How many circles can you draw through three distinct points A, B and C on a plane? Is there always at least one such circle? Not necessarily! What if A, B and C lie on a straight line, i.e., are collinear? Can you explain why, in this case, there is no circle through A, B and C?
- Let us assume that A, B and C are not collinear. Is there always a circle passing through A, B and C? Can there be more than one circle through A, B and C?