Problem Solving

Please help.
Thanks.

For any point (x, y), √(x² + y²) = the distance between (x, y) and the origin.

W = (a, b)
X = (c, d)
Since we are told that a² + b² = c² + d², we get:
√(a² + b²) = √(c² + d)
(distance between W and the origin) = (distance between X and the origin).

The same reasoning can be applied to Y and Z.
Thus:
(distance between Y and the origin) = (distance between Z and the origin).

Question: What is XZ - WY?

XZ = (distance between X and the origin) + (distance between Z and the origin)
WY = (distance between W and the origin) + (distance between Y and the origin)
Since the values in red are equal and the values in blue are equal, XZ = WY.
Thus:
XZ - WY = 0.

The correct answer is C.

Thanks Mitch!

didieravoaka wrote:Thanks Mitch!

You could also think of this visually. These four points lie on one circle, so XZ and WY are both diameters of that circle, and hence have equal length, making their difference zero.

Thanks matt! I thought considering such problems visually was not advised. Though, this is how I got the correct answer when I was taking that test.

didieravoaka wrote:Thanks matt! I thought considering such problems visually was not advised. Though, this is how I got the correct answer when I was taking that test.

Excellent!!

The more geometric the solution to a geometry problem, the more elegant it is, I think. I have to grind these out with algebra sometimes myself, but I always feel dirty doing it.

Ahahahaha
You're definitely right.
Thanks again.