Question

The lengths of the sides of a right triangle are consecutive even integers. Find these lengths. (Hint: Use the Pythagorean theorem.)

Answer 1

`6,8,10`

Explanation:

`\mathrm{Let\ the\ three\ consecutive\ even\ integers\ be\ 2x,\ 2x+2\ and\ 2x+4.}\\\mathrm{Then,}\\\mathrm{From\ pythagoras\ theorem,}\\\mathrm{(2x+4)^2=(2x+2)^2+(2x)^2}\\\mathrm{or,\ 4x^2+16x+16=4x^2+8x+4+4x^2}\\\mathrm{or,\ 16x+16=8x+4+4x^2}\\\mathrm{or,\ 4x+4=2x+1+x^2}\\\mathrm{or,\ x^2-2x-3=0}\\\mathrm{or,\ x^2-3x+x-3=0}\\\mathrm{or,\ x(x-3)+1(x-3)=0}\\\mathrm{i.e.\ x=3\ or\ -1.}\\\mathrm{Negative\ value\ of\ x\ is\ neglected\ since\ length\ cannot\ be\ negative.}\\\mathrm{\therefore x=3}`

`\mathrm{So,\ 2x+4=2(3)+4=10}\\\mathrm{2x+2=2(3)+2=8}\\\mathrm{2x=2(3)=6}\\\mathrm{Hence,\ 6,8\ and\ 10\ are\ the\ required\ lengths\ of\ triangle.}`