Circle

A and C must be on the two ends of the diameter when x getting maximum.ㄥABC = 90° (ㄥS in the semi-circle)SoAB² + BC² = AC² = 10² = 100 .......(1)AB + BC = x
BC = x - AB .......(2)Sub (2) into (1) :AB² + (x - AB)² = 1002AB² - 2x AB + x² - 100 = 0△ = (2x)² - 4(2)(x² - 100) ≥ 0 for AB is real ,4x² - 8x² + 800 ≥ 0x² - 200 ≤ 0(x - 10√2)(x + 10√2) ≤ 0- 10√2 ≤ x ≤ 10√2The maximum value of x = 10√2 cm

2010-10-20 19:14:43 補充：
001 :

x is the sum of the length of the line segments AB and BC.

Not arc segments.

the length will be longest if A and C is on the corner

http://img221.imageshack.us/img221/8900/200.png

the answer=10*3.14
=31.4cm