Lenny Conundrum?

Question

For all you Math brainiacs-Can you please help me with this weeks Lenny Conundrum? (TY)
Ok here it is.....The Petpet Supplies shop is 655 metres from the Smoothie Store, 393 Metres from the Defence Magic shop, and 314 metres from the Grooming Parlour. The Defence Magic shop is 236 metres from the Grooming Parlour and 524 metres from the Smoothie Store.
Assuming the distance between the Grooming Parlour and the Smoothie Store is less than the distance between the Petpet Supplies shop and the Smoothie Store, what is the distance between the Grooming Parlour and the Smoothie Store? Please round to the nearest metre and show me how you arrived at the answer.

Answer 1

This sounds like trigonometry, set it up in the coordinate plane and solve.

I get 364 meters.

Label Petpet at the origin (0,0), Smoothie at (655,0) on the +x axis. Let Defence be at (a,b) on the circle a^2+b^2=393^2 and let Grooming be at (x,y) on the circle x^2+y^2=314^2. Then the other distance are DG=234 and DS=524 which produce two more equations (a-655)^2 +b^2=524^2, (a-x)^2+ (b-y)^2=236^2 and then you solve these. One way (by hand) is too expand and eliminate. The other way is technology (like the Maple output below):

> soln:=fsolve({a^2+b^2=393^2,x^2+y^2=314^2,(a-655)^2+b^2=524^2,(a-x)^2+(b-y)^2=236^2},{x,y,a,b});
soln := {a = 235.8000000, b = -314.4000000, x = 301.4958441,
y = -87.72830775}
> subs(soln,((x-655)^2+y^2)^.5);
364.2271876

Not trigonometry just algebra. I bet you could use trig if you wanted to.

Answer 2

Angle A is the angle between 393 and 236
Angle B is the angle between 393 and 524
Angle C is the angle between 236 and 524

3142=3932+2362-2(393)(236)cosA
arccos(0.60135528528917065597101824298098) = 0.9256 radians *180/pi =
Angle A = 34.455119834955038445497247225593 degrees

6552=3932+5242-2(393)(524)cosB
arccos(0) = pi/2 radians *180/pi =
Angle B = 90 degrees
Angle C = Angle B - Angle A = 55.54488016504496155450275277441 degrees

X2=2362+5242-2(236)(524)cosC
X = 436.28383213555175413619346662222 meters (... and round)