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?
Step-by-step solution
Idea: When the boundaries overlap, every point P of the edge lands on another edge point P′, and the crease is the perpendicular bisector of PP′. The centre O is equidistant from P and P′ (both are radii), so O lies on that perpendicular bisector, which is the crease.
- Take any point P on the edge. When the paper is folded with the boundaries overlapping, P lands on some point P′ of the edge.½ mark
- In a fold, the crease is the perpendicular bisector of the segment joining a point to the point it lands on. So the crease is the perpendicular bisector of PP′.½ mark
- The centre O is at the same distance from P and P′, since OP = OP′ = radius. Every point equidistant from P and P′ lies on the perpendicular bisector of PP′.½ mark
- So O lies on the crease. The crease is a line through the centre, so the part of it inside the circle is a diameter.½ mark
Check: Fold a second time in another direction: the two creases cross at one point, and a ruler shows this point is the same distance from every point of the edge.
Answer to write in the exam
On folding, point P of the edge falls on point P′ of the edge.
Crease = perpendicular bisector of PP′ (property of a fold)
OP = OP′ (radii) ⇒ O lies on the perpendicular bisector of PP′
∴ Yes; the crease passes through the centre O, i.e. it is a diameter.
Common mistakes that cost marks
- Saying “yes, because it looks like it”. The reason is that the centre is equidistant from P and its folded image P′, so it lies on the perpendicular bisector of PP′, which is the crease.
- Thinking any chord is a line of symmetry. Only chords through the centre (diameters) are.
- Confusing a line of reflection symmetry with rotational symmetry. Folding tests reflection; turning tests rotation.
How this can come in the exam
How many lines of reflection symmetry does a circle have?
- 1
- 2
- 4
- Infinitely many
Show answer
(D) Infinitely many
Every diameter is a line of reflection symmetry, and a circle has infinitely many diameters.
Assertion (A): Every line of reflection symmetry of a circle passes through its centre.
Reason (R): The centre of a circle is equidistant from every point on the circle.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true, but R is false.
- A is false, but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
Both are true. A line of symmetry is the perpendicular bisector of PP′ for each edge point P and its image P′; since the centre is equidistant from P and P′ (R), it lies on that line. So R explains A.
Try one yourself
Is a chord that does not pass through the centre a line of symmetry of the circle? Give a reason.
Show answer
No. If we fold along such a chord, the smaller piece of the circle does not cover the larger piece, so the boundaries do not overlap. Only diameters are lines of symmetry.
More questions like this
- 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?
- Draw ΔABC with AB = 5 cm, ∠A = 70° and ∠B = 60°. Draw the circumcircle of ΔABC. Is the centre inside or outside the triangle?