Geometry: An Interactive Journey to Mastery

Dido’s Problem

Lesson 32

Topics
x The legend of Dido.
x The isoperimetric problem and its solution.
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x isoperimetric problem7KHFKDOOHQJHRIGHWHUPLQLQJZKLFK¿JXUHLQWKHSODQHKDVWKHJUHDWHVWDUHD
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Result
x Of all shapes with a given perimeter, the circle encloses the maximal area.
Summary
In this lesson, we use the legend of Dido to motivate a famous problem in geometry: the isoperimetric
problem. We prove what the answer to the problem must be—if you believe that the problem has an answer in
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Example 1
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Solution
Suppose that the base of the triangles under consideration is the
horizontal line segment FG.)RUPLQJDWULDQJOHZLWKSHULPHWHURI
length L requires locating a point P such that FP + PG equals the
¿[HGYDOXHLíFG. As we saw in Lesson 29, the locus of all
possible such points PLVDQHOOLSVHͼ௘6HHFigure 32.1௘ͽ

P

FG

Figure 32.1