Equidistant Points

The “House” Construction Problem

Imagine you need to draw a roof for a house. You have the base $BC$, and you need a peak point $A$ such that $AB = 5$ cm and $AC = 5$ cm.

The Concept

How do we find a point that is exactly 5 cm from $B$ and 5 cm from $C$?

1. Locus from B: All points 5 cm from $B$ lie on a circle centered at $B$ with radius 5 cm.
2. Locus from C: All points 5 cm from $C$ lie on a circle centered at $C$ with radius 5 cm.
3. Intersection: The point where these two circles meet satisfies both conditions.

Steps

1. Draw the base line (e.g., $BC$).
2. With $B$ as center and radius 5 cm, draw an arc.
3. With $C$ as center and radius 5 cm, draw an arc intersecting the first one.
4. Mark the intersection point $A$.
5. Join $AB$ and $AC$.

This method creates an Isosceles Triangle (or Equilateral if the base is also 5 cm).