An isosceles triangle is drawn so that it has the same area as the above square (i.e. x), and with two sides that are equal to the square root of x (henceforth dubbed y). What is the length of the third side?
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Let the length of the third side be 2z, and the length of the perpendicular to the third side be p.
Each side of the isosceles triangle will be the hypotenuse of a triangle whose other sides are p and z.
So, z2 + p2 = y2 = x
Now, area of the isosceles triangle
= 1/2 * Perpendicular * Third side
= (1/2) * p * 2z
= zp
Since the areas are equal, zp = x
So, z2 + p2 = pz
z2 - zp + p2 = 0
The above is a quadratic equation in z, whose discriminant = (p)2 - 4 * 1 * (p2) = -3p2 < 0
So, the value of z is not real.
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