Geometric twins

A symbol on a signboard needs to be recreated on another board. How do we do it? One way is to trace its outline on tracing paper — but that is difficult for a big symbol. What else can we do?

Take MEASUREMENTS that would let us recreate the figure exactly. But which measurements? Name the corner points A, B and C, and start with the two arm lengths AB and BC.

Are those enough? Suppose AB = 4 cm and BC = 8 cm. Draw it — and then draw it again with the arms at a different opening. Several quite different symbols have exactly those two lengths. So two lengths are not enough.

What else would help? The measure of ∠ABC — the angle between the two arms. Given AB = 4 cm, BC = 8 cm and ∠ABC = 80°, the shape and size are fixed, and anyone can draw the exact replica.

Figures that are exact copies of each other, having the same shape AND the same size, are said to be CONGRUENT (in Hindi, सर्वांगसम) — geometric twins. Congruent figures can be superimposed exactly, one over the other. Trace one on tracing paper, lay it over the other, and it fits.

One thing to watch while checking. A figure may be ROTATED or FLIPPED before superimposing it on the other. A pair that looks quite different on the page may fit perfectly once one of them is turned round or turned over — and such a pair is congruent. Your two hands are an example: they look reversed, but press them palm to palm and they match, because one has been flipped over.

So the question 'are these congruent?' becomes 'what measurements do I need in order to be sure?' For a circle, the radius alone is enough. For a rectangle, the length and the breadth. The rest of the chapter asks it for triangles.