標題:
Arithmetic sequence question
發問:
If the perimeter of the right-angled triangle (x is the opposite side, y is the adjacent side and z is the hypotenuse) is 42cm, and the values of x,y and z form an arithmetic sequence, find the lengths of the sides of the triangle. (please show detailed steps)
最佳解答:
x, y and z form an arithmetic sequence. Let a and d are the first term and the common difference respectively. x = a cm y = (a + d) cm z = (a + 2d) cm Perimeter: a + (a + d) + (a + 2d) = 42 ...... (1) Pythagoras theorem: a2 + (a + d)2 = (a + 2d)2 ...... (2) (1): a + a + d + a + 2d = 42 3a + 3d = 42 a + d = 14 d = 14 - a ...... (3) (2): a2 + (a2 + 2ad + d2) = a2 + 4ad + 4d2 a2 - 4ad - 3d2 = 0 ...... (4) Put (3) into (4): a2 - 4a(14 - a) + 3(14 - a)2 = 0 a2 - 4a(14 - a) + 3(196 - 28a + a2) = 0 a2 - 56a + 4a2 + 588 - 84a + 3a2 = 0 8a2 - 140a + 588 = 0 2a2 - 35a + 147 = 0 (2a - 21)(a - 14) = 0 a = 21/2 or a = 14 (rejected) a = 10.5 Put a = 10.5 into (3) d = 14 - 10.5 d = 3.5 x = a cm x = 10.5 cm y = (a + d) cm y = 14 cm z = (a + 2d) cm z = 17.5 cm Ans: The sides of the triangle are 10.5 cm, 14 cm and 17.5 cm respectively. =
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