Henry Ernest Dudeney/Modern Puzzles/133 - A Problem for Surveyors/Solution

Modern Puzzles by Henry Ernest Dudeney: $133$

A Problem for Surveyors

A man bought a little field, and here is a scale map that was given to me.

I asked my surveyor to tell me the area of the field,

but he said it was impossible without some further measurements;

the mere length of one side, $7$ rods, was insufficient.

What was his surprise when I showed him in about two minutes what was the area!

Can you tell how it is to be done?

Solution

The construction given below takes advantage of the fact that $\angle AFG$ is a right angle.

Produce $GA$.

Construct $AC$.

Construct a straight line through $B$ parallel to $AC$ to meet $GA$ at $1$.

Construct $1D$.

Construct a straight line through $C$ parallel to $1D$ to meet $GA$ at $2$.

Construct $2E$.

Construct a straight line through $D$ parallel to $2E$ to meet $GA$ at $3$.

Construct $3F$.

Construct a straight line through $E$ parallel to $3F$ to meet $GA$ at $4$.

Construct $4G$.

Then $\triangle G4F$ is equal in area to the field $ABCDEFG$.

Proof

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Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $133$. -- A Problem for Surveyors
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $263$. A Problem for Surveyors