The length, in cm, of the sides of a right angled triangle are x, (x+2) and (x+1) where x > 0. Find , in cm, the length of its hypotenuse
The longest side of a right angle triangle is the hypotenuse.
\((x + 2)^{2} = x^{2} + (x + 1)^{2}\)
\(x^2 + 4x + 4 = x^2 + x^2 + 2x + 1\)
\(2x^{2} - x^{2} + 2x - 4x + 1 - 4 = 0\)
\(x^{2} - 2x - 3 = 0\)
\(x^{2} - 3x + x - 3 = 0 \implies x(x - 3) + 1(x - 3) = 0\)
\((x - 3)(x + 1) = 0 \implies \text{x = 3 or -1}\)
\(x > 0 \implies x = 3\)
The longest side = 3 + 2 = 5.
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