Question 1196592
<br>
The given information defines a picture consisting of two right triangles sharing one leg of length 24 and with the other legs of lengths x and x+11; and the hypotenuse of the larger triangle is 5 longer than the hypotenuse of the smaller triangle.<br>
{{{drawing(400,400,-2,20,-2,30
,line(0,0,20,0),line(0,0,0,26),line(0,24,7,0),line(0,24,18,0)
,locate(3,0,x),locate(10,0,11),locate(.5,12,24),locate(0,0,O),locate(.5,25,A),locate(7,0,B),locate(18,0,C)
)}}}<br>
The requirement is that AC is 5 longer than AB:<br>
{{{sqrt(24^2+(x+11)^2)=sqrt(24^2+x^2)+5}}}<br>
Solving that equation algebraically is ugly; using a graphing calculator would be much easier.<br>
But since the numbers in the problem are whole numbers, we can guess the solution knowing something about Pythagorean Triples.<br>
A relatively well-known Pythagorean Triple is 7-24-25. So if x is 7 and AB is 25, then x+11 is 18 and AC is 30; and 18-24-30 is also a Pythagorean Triple.<br>
So, whether you solve the problem algebraically, or using a graphing calculator, or by logical trial and error, the lengths of the two wires are 25 feet and 30 feet.<br>