# Please help me solve this problem: Ahmed has half 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? Thank you. Ahmed has half 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.

The two decide to share the information available in their maps. They get to Castle Rock, and walk x paces to the north and then 2x + 4 paces to the east. This position is 2x + 6 paces from Castle Rock. Using the Pythagorean Theorem gives:

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

=> x^2 + 4x^2 + 16 + 16x = 4x^2 + 36 + 24x

=> x^2  =  20 + 8x

=> x^2 - 8x - 20 = 0

=> x^2 - 10x + 2x - 10 = 0

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

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

=> x = -2 and x = 10

Ignoring the negative solution of the equation, x = -2, we get x = 10

The value of x determined from the information provided by both their maps is 10.

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