I can't figure this word problem out.

Question

Instead of walking along two sides of a rectangular field, Fran took a shortcut along the diagonal, thus saving distance equal to half the length of the longer side. Find the length of the long side of the field, given that the length of the short side is 156 meters.

Answer 1

The length of the diagonal is equal to the length of the long side plus the length of the short side minus half the length of the long side because you said Fran saved a distance equal to half the length of the long side.

x=length of the long side

156 m=length of the short side

156+x-0.5x=length of the diagonal

using the Pythagorean Theorem

(156+x-0.5x)²=156²+x²

(156+0.5x)²=156²+x²

156²+156x+0.25x²=156²+x²

cancel out both of the 156² because there is one on each side

156x+0.25x²=x²

x²-0.25x²-156x=0

0.75x²-156x=0

divide both sides by 0.75

x²-208x=0

factor

x(x-208)=0

x-208=0

x=208 m is the length of the long side

the diagonal is 156+208-(1/2)(208)=364-104=260

260²=208²+156² and...

the length plus the width is 208+156=364

260 is 364-(1/2)208), 260=364-104, 260=260

again, the length of the long side is 208 m