Geometry: An Interactive Journey to Mastery

Lesson 32: Dido’s Problem

Because the area of the triangle is^12 × FG × height and FGLVD¿[HGYDOXHWKHWULDQJOHRIPD[LPDODUHD

occurs when P is at the position of maximal height. This occurs when P is directly above the midpoint of FG

and the triangle FPGLVLVRVFHOHVͼ௘7KLVFODLPLVLQWXLWLYHO\FOHDU&DQ\RXSURYHLW":KLFKYDOXHRIx gives the

largest possible value for y in the equation abx^222  y^21 RIDQHOOLSVH"௘ͽ

Thus, of all triangles with FG as its base, the isosceles triangle has the largest area.

Example 2

Of all triangles with perimeter 15 inches, which encloses the largest area?

Solution

$Q\WULDQJOHWKDWLVQRWHTXLODWHUDOFDQQRWEHWKHDQVZHU)RUH[DPSOHZHFDQ

increase the area of the triangle shown in Figure 32.2 by moving P along the

SDWKRIDQHOOLSVHͼ௘UHJDUGLQJQRDVD¿[HGEDVHDNLQWR([DPSOH௘ͽ7KLV

triangle fails to have maximal area, and the same argument shows that the

area of any triangle that is not equilateral can be increased and, therefore, fails

to have maximal area, too.

So, this leaves the 5-5-5 equilateral triangle as the only possible candidate to answer this question.

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Study Tip

x This lesson is purely optional. There are no recommended study tips for this lesson other than to enjoy

the lesson and let the thinking about it strengthen your understanding of geometry as a whole.

Pitfall

x Don’t forget to have fun in your thinking of mathematics. This is a fun topic.

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Figure 32.2