To Prove: The perimeter of A < the perimeter of B. Proof : Construct regular polygon s, similar to B and isoperimetric with 4.

Then A > 8 (477), but A≈ B (?). .'. B>S (?) (Ax. 6).
Hence, the perimeter of B > perimeter of s (390).
But, the perimeter of s = perimeter of A (?).

.. perimeter of B > perimeter of ▲ (Ax. 6).
That is, the perimeter of A< the perimeter of B.

Q.E.D. 480. THEOREM. Of all equivalent plane figures the circle has the minimum perimeter.

ORIGINAL EXERCISES

1. Of all equivalent parallelograms having equal bases the rectangle has the minimum perimeter.

2. Of all lines drawn between two given parallels (terminating both ways in the parallels), which is the minimum? Prove.

3. Of all straight lines that can be drawn on the ceiling of a room 12 feet long and 9 feet wide, what is the length of the maximum?

4. Find the areas of an equilateral triangle, a square, a regular hexagon, and a circle, the perimeter of each being 264 inches. maximum? What theorem does this exercise illustrate ?

Which is

5. Find the perimeters of an equilateral triangle, a square, a regular hexagon, and a circle, if the area of each is 1386 square feet. Which perimeter is the minimum? What theorem does this exercise illustrate? 6. Of isoperimetric rectangles which is maximum? 7. To divide a given line into two parts such that their product (rectangle) is maximum.

8. Of all equivalent triangles having the same base the isosceles triangle has the minimum perimeter.

To Prove: The perimeter of ▲ ABC > the perimeter of

AB'C.

Proof: ADAB' + B'C'; etc.

A

9. Of all rectangles inscribed in a circle which is maximum? Prove. 10. Of all rectangles inscribed in a semicircle which is maximum? Prove.

11. Of all equivalent rectangles, the square has the minimum perimeter.

12. Of all triangles having a given base and a given vertex-angle, the isosceles triangle has the maximum perimeter.

13. Of all triangles having a given altitude and a given vertex-angle, the isosceles triangle is the minimum.

14. Of all triangles that can be inscribed in a given circle the equilateral triangle has the maximum area and the maximum perimeter.

15. The cross section of a bee's cell is a regular hexagon. Would this be the most economical for the bee, if one cell in a hive were all he were to fill (that is, would he use the least wax)? Considering also the adjoining cells, does the form of the regular hexagon require the least wax? Explain. Does it also permit the storing of the most honey? Why?

16. Prove that a regular hexagon is greater than an isoperimetric square, by the method employed in 477.

17. Answer the questions of exercise 65 on page 243, without any computation. Give reasons.

18. Compare the areas of the figures mentioned in exercise 66, page 243, without performing any computation.