Question 1

We can find points that are equally far from the ends of a line segment.

Step 1 — Drawing the line segment

Draw a straight line segment. Let us call its ends X and Y.

Step 2 — Finding a point above

We need a point equally far from X and Y. Open your compass to a distance. This distance must be more than half of XY. Let us call this distance k. Place the compass point on X. Draw an arc above the line XY. Keep the compass open to the same distance k. Place the compass point on Y. Draw another arc above the line XY. These two arcs will cross each other. Let us call this crossing point A. So, the distance from X to A is k. The distance from Y to A is also k. This means AX = AY.

Step 3 — Finding a point below

Now we need another point equally far from X and Y. We can use a different distance for this. Open your compass to a new distance. This new distance must also be more than half of XY. Let us call this new distance k'. Place the compass point on X. Draw an arc below the line XY. Keep the compass open to the same distance k'. Place the compass point on Y. Draw another arc below the line XY. These two arcs will cross each other. Let us call this crossing point B. So, the distance from X to B is k'. The distance from Y to B is also k'. This means BX = BY. Notice that k and k' can be different.

Step 4 — Drawing the bisector

We have two points, A and B. Point A is equally far from X and Y. Point B is equally far from X and Y. Any point that is equally far from the ends of a line segment lies on its perpendicular bisector. So, both A and B are on the perpendicular bisector of XY. Draw a straight line connecting point A and point B. This line AB is the perpendicular bisector of XY. It cuts XY into two equal parts. It also makes a 90-degree angle with XY.

Step 5 — Justifying the radii

For point A, we used radius k from both X and Y. So, AX = AY = k. For point B, we used radius k' from both X and Y. So, BX = BY = k'. The line AB connects all points that are equally far from X and Y. Since A is equally far from X and Y, it is on the bisector. Since B is equally far from X and Y, it is on the bisector. We only need two such points to draw the line. The radii k and k' do not need to be the same. They just need to be large enough to make the arcs cross.

Answer

(i) No, it is not necessary to use the same radius for the arcs above and below XY. (ii) For the arcs above XY, the radius from X must be the same as the radius from Y. (iii) For the arcs below XY, the radius from X must be the same as the radius from Y. (iv) The radius used for the upper arcs can be different from the radius used for the lower arcs.