Buried treasure. Ahmed has half of a treasure map,which indicates that the treasure is buried in the desert 2x + 6 paces from Castle Rock. Vanessa has the other half of the map. Her half indicates that to find the treasure, one must get to Castle Rock, walk x paces to the north, and then walk 2x + 4 paces to the east. If they share their information, then they can find x and save a lot of digging. What is x?

Thomas is right.

From Castle rock, the northward leg of the right triangle is x paces. The eastward leg from there is (2x+4) paces, and the straight line distance (the hypotenuse) is (2x+6) paces. We can make a quadratic equation with this information.

As Thomas pointed out, the Pythagorean theorem states that the square of the hypotenuse is the sum of the squares of the legs:

a^{2} + b^{2} = c^{2}

x^{2} + (2x+4)^{2} = (2x+6)^{2}

Expand the squares and combine terms.

x^{2} + 4x^{2} + 16x + 16 = 4x^{2} + 24x + 36

x^{2} + 4x^{2} - 4x^{2} + 16x - 24x + 16 - 36 = 0

x^{2} - 8x - 20 = 0

We can factor this to get:

(x-10)(x+2) = 0

So x could be 10 or -2. Negative 2 doesn't make sense in this situation, so:

**x = 10.**