Jamuna has a circular piece of paper. She is trying to locate its centre. Amina gives her a suggestion. She follows the instructions and is thrilled to find that it works. Can you guess what Amina told her?
Step-by-step solution
Idea: Any fold that makes the edge of a paper circle lie exactly on itself is a line of symmetry, and every line of symmetry of a circle is a diameter, so it passes through the centre. Two different diameters can meet at only one point: the centre.
- Step 1 (first fold): Fold the paper circle in half so that the two halves of the edge lie exactly on top of each other. Open it. The crease is a line of symmetry of the circle.½ mark
- Why the crease passes through the centre: the centre is the one point at the same distance from every point of the edge. When the two halves match, the centre must land on itself, so it lies on the crease. The crease is a diameter.1 mark
- Step 2 (second fold): Fold the circle in half again along a different direction and open it. This crease is another diameter.½ mark
- Step 3: Both creases pass through the centre, and two different straight lines cross at only one point. So the point where the creases cross is the centre.1 mark
- Another way (if the paper must not be folded): draw any two chords, draw the perpendicular bisector of each with ruler and compass; they meet at the centre, because the centre is equidistant from the ends of every chord.
Check: Measure from the crossing point to several points on the edge with a ruler: all the distances come out the same (the radius).
Answer to write in the exam
Fold the paper circle so that the edges overlap exactly; open it. Crease 1 = line of symmetry = a diameter (passes through the centre).
Fold again along a different direction; open it. Crease 2 = another diameter.
Two distinct lines meet in exactly one point.
∴ The point where the two creases cross is the centre.
Common mistakes that cost marks
- Folding only once. One crease is a diameter, but the centre could be anywhere along it; a second crease is needed to fix the point.
- Making a fold where the edges do not overlap exactly. Such a crease is just a chord, not a diameter, and does not pass through the centre.
- Folding twice along the same line: the two creases coincide and do not give a single crossing point.
How this can come in the exam
A paper circle is folded so that its edge lies exactly on itself, opened, and folded again the same way along a different line. The two creases meet at
- a point on the circle
- the centre of the circle
- the midpoint of a radius
- a different point each time
Show answer
(B) the centre of the circle
Each such crease is a line of symmetry, that is, a diameter. Two different diameters meet only at the centre.
A circle is drawn on a sheet of paper but its centre is not marked. Using only a ruler and compasses, how would you find the centre? Give a reason.
Show answer
Mark any three points A, B, C on the circle. Construct the perpendicular bisectors of AB and BC (1 mark). Every point on the perpendicular bisector of AB is equidistant from A and B, and the centre is equidistant from A, B and C, so it lies on both bisectors; their point of intersection is the centre (1 mark).Try one yourself
Raju folds a paper circle so that the edges do not quite overlap and gets a crease. Will this crease pass through the centre? Explain.
Show answer
No. Only a fold in which the two halves of the edge match exactly is a line of symmetry (a diameter). Any other crease is just a chord that misses the centre.
More questions like this
- 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?
- 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?