Question 3

We will explore if using the same radius for arcs helps us find special points.

Step 1 — Drawing a line segment

Let us draw a line segment. We will call its endpoints X and Y.

Step 2 — Drawing arcs with unequal radii

Let us open our compass to a length. We will call this length k. With X as the center, we draw an arc. Now, let us open our compass to a different length. We will call this length k'. With Y as the center, we draw another arc. These two arcs cross each other at a point. Let us call this point A. The distance from X to A is k. The distance from Y to A is k'. Since k is not equal to k', the distance XA is not equal to YA.

Step 3 — Understanding the perpendicular bisector

A perpendicular bisector is a special line. It cuts a line segment exactly in half. It also forms a right angle with the segment. Every point on this special line is equally far from the segment's ends. For a point R on the perpendicular bisector of XY, RX must equal RY.

Step 4 — Checking point A

From Step 2, we know that XA is not equal to YA. This means point A is not equally far from X and Y. So, point A cannot be on the perpendicular bisector of XY. Using unequal radii does not help us find points for this special line.

Step 5 — Drawing arcs with equal radii

Let us draw the line segment XY again. We open our compass to a length. We will call this length r. This length r must be more than half of XY. With X as the center, we draw an arc. Now, with Y as the center, we use the same length r. We draw another arc. These two arcs cross each other at two points. Let us call these points P and Q. The distance from X to P is r. The distance from Y to P is also r. So, PX equals PY. The distance from X to Q is r. The distance from Y to Q is also r. So, QX equals QY. Both P and Q are equally far from X and Y.

Step 6 — Justifying the necessity of equal radii

We want to construct the perpendicular bisector of XY. This line needs points that are equally far from X and Y. When we used unequal radii, point A was not equally far from X and Y. So, point A cannot be on the perpendicular bisector. When we used equal radii, points P and Q are equally far from X and Y. The line joining P and Q will be the perpendicular bisector. Therefore, to find points for the perpendicular bisector, we must use the same radii.

Answer

(i) No, it is not necessary to use the same radii for any pair of intersecting arcs. (ii) However, to construct a perpendicular bisector, it is necessary to use the same radii for both arcs. (iii) This is because points on a perpendicular bisector must be equidistant from the segment's endpoints.