## Square Lot

Al and Curt live at opposite corners of a square lot; Beth and Dawn live at the other two corners. They all carry water from a spring $S$ located within the lot. The spring is $50$ feet from Al, $30$ feet from Beth, and $40$ feet from Curt. How far is the spring from Dawn?
Source: NCTM Mathematics Teacher 2008

SOLUTION

$a^2+b^2=x^2\qquad\:\: (1)$
$a^2+c^2=50^2\qquad (2)$
$c^2+d^2=30^2\qquad (3)$
$b^2+d^2=40^2\qquad (4)$
Subtract Eq. (2) from Eq. (1)
$b^2-c^2=x^2-50^2\qquad\:\: (5)$
Subtract Eq. (3) from Eq. (4)
$b^2-c^2=40^2-30^2\qquad (6)$
Compare Eq. (5) and Eq. (6)
$x^2-50^2=40^2-30^2$
$x^2=40^2-30^2+50^2$
$=3200$
$x=56.6$ feet

Answer: $56.6$ feet