While constructing one pair of intersecting arcs, is it necessary that we use the same radii for both of them ? Explore this through construction, and then justify your answer.

Question

Homi · Accepted Answer

Understand the Question

Any two arcs with centers $X$ and $Y$ can intersect as long as their radii satisfy the triangle inequality ( $k + k' \ge XY$ ), regardless of whether the radii are equal or unequal.
However, for specific geometric constructions like a perpendicular bisector, every point on the bisector must be equidistant from both endpoints ( $X$ and $Y$ ).
Therefore, whether equal radii are necessary depends on the purpose of the construction.

Step 1 · Construct Arcs with Unequal Radii

Draw a line segment $XY$ . Diagram 1

With center $X$ , draw an arc of radius $k$ . With center $Y$ , draw an arc of a different radius $k'$ ( $k \neq k'$ ). Let these arcs intersect at point $A$ . Diagram 2

From the construction:

XA = k, \quad YA = k'

XA \neq YA

The arcs intersect successfully at $A$ , but point $A$ is not equidistant from $X$ and $Y$ . Since any point on the perpendicular bisector must be equidistant from both endpoints ( $RX = RY$ ), point $A$ does not lie on the perpendicular bisector of $XY$ .

Step 2 · Construct Arcs with Equal Radii

Draw arcs from centers $X$ and $Y$ using the same radius $r$ , where $r > \dfrac{1}{2}XY$ . Let them intersect at points $P$ and $Q$ . Diagram 3

From the construction:

\begin{aligned} PX &= PY = r \\[0.4em] QX &= QY = r \end{aligned}

Since points $P$ and $Q$ are equidistant from endpoints $X$ and $Y$ , the line joining $P$ and $Q$ forms the perpendicular bisector of $XY$ .

Answer

No, it is not necessary to use equal radii merely to make two arcs intersect. However, to construct a perpendicular bisector, equal radii are mandatory so that the points of intersection remain equidistant from both endpoints.

Common Mistakes

Assuming Equal Radii Are Always Required: Two arcs can intersect with completely different radii as long as the sum of their radii is greater than the distance between their centers ( $k + k' > XY$ ).
Radius Too Small in Bisector Construction: For arcs with equal radii $r$ to intersect, $r$ must be strictly greater than half the segment length ( $r > \frac{1}{2}XY$ ); otherwise, the arcs will never meet.

Question 3

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