consider the expression (√a+√b)^2 where a and b are not perfect squares find 2 sets of the values of a and b such that the value of the expression is a rational no
ABCD is a square and BCE is an equilateral triangle may thinks that the ratio of the area of ABECD to the area of triangle BCE must be a constant do u agree with her?EC =AB explain u ans briefly
connee
2010-08-10 11:17 pm
回答 (1)
2010-08-10 11:44 pm
✔ 最佳答案
(√a+√b)^2= a + 2√a√b + b= a + b + 2√(ab)The expression is a rational no, therefore √ab must be rational ,ab is a perfect square , Set a = p^3 , b = p ,ab = p^3 * p = p^4 is a perfect square ,So(√a+√b)^2= (√p^3 + √p)^2 is rational.Putting p = 3 , (√27 + √3)^2 = 48 is rational while 27 and 3 are not perfect squares. Agree.Let x be the side length of the square and the triangle BCE ,ABECD : △BCE = (Square - △BCE) : △BCE = (x^2 - (1/2)(sin60)x^2) : [(1/2)(sin60)x^2]= (1 - (1/2)(sin60)) : [(1/2)(sin60)] which is a constant.EC = BC as they are sides of a equilateral triangle ,BC = AB as they are sides of a square ,So EC = AB
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