Question

Judy planted vegetables on a piece of land. The land is in the right-angled triangular shape. Given the longest side of the land is y metre. The other two sides of the land are 2x metre and (2x+6) metre respectively. He fenced the land with 72 metres of barbed wire. Find the length, in metre , of each side of the land.

Answer 1

We know that the perimeter is 72 meters, so we have

$y+2x+(2x+6)=72 \iff y+4x+6=72 \iff y=-4x+66$

Now we can use the pythagorean theorem: the sum of the squares of the legs is the square of the hypotenuse:

$(2x)^2+(2x+6)^2 = y^2 \iff (2x)^2+(2x+6)^2 = (-4x+66)^2$

Expand the squares to get

$8 x^2 + 24 x + 36 = 16 x^2 - 528 x + 4356$

Bring everything to the right hand side to get

$8 x^2 - 552 x + 4320 = 0$

Divide both sides by 8:

$x^2 - 69x + 540 = 0$

This equation has solutions $x=9,\ x=60$

This solutions would yield the following side lengths:

$x=9\implies\begin{cases} 2x = 18\\2x+6=24\\-4x+66=30\end{cases}$

$x=60\implies\begin{cases} 2x = 120\\2x+6=126\\-4x+66=-174\end{cases}$

Since we can't have negative side lengths, we can only accept the solution x=9.

In fact, you can check that 18,24,30 is a pythagorean triplet, i.e.

$18^2+24^2=30^2$