Mary told her grandmother that the sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. If each of the two equal sides of the isosceles triangle is represented by the letter a, find the area of the isosceles triangle in terms of a.

The area of such a triangle is zero.

Expert Answers

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Denote the length of the third side of this isosceles triangle as `b . ` Then, the given condition "the sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side" means that

`1) sqrt ( a ) + sqrt ( a ) = sqrt ( b ) ,`

`2) sqrt ( a ) + sqrt ( b ) = sqrt ( a ) .`

The second condition alone implies `sqrt ( b ) = 0 , ` so `b = 0 . ` If some side of a triangle has zero length, the area of the triangle is zero.

But even the first condition alone leads to the same conclusion. Indeed, it means `2 sqrt ( a ) = sqrt ( b ) , ` which gives `4 a = b ` when squared. But a triangle cannot have sides `a , ` `a ` and `4 a ` because it violates the triangle inequality: `a + a lt= 4 a .` The only option for this to produce at least a degenerate triangle is to have `a = 0 . ` Then `b = 0 , ` too, and the area is zero.

Actually, because of the triangle inequality a condition `a^p + b^p = c^p ` can produce a valid triangle only for `p gt 1 , ` but here `p = 1 / 2 lt 1 .`

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