Solve using the Theorem theory

Question

Can you help me solve this problem using the Pythagorean Theorem

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?

Let a = x, and b = 2x + 4 so that c = 2x + 6

Thanks for your help

Answer 1

First of all, you need to prove that the distances are all part of a right triangle.  Cardinal directions such as north and east are perpendicular to each other on a compass.  This proves that the angle formed between walk north and east is 90 degrees.  As x and 2x+4 intersect at this 90 degree angle then it must be deduced that 2x+6 is the length of the hypotenuse. This is where we derive that in the Pythagorean theorem c is always the long hypotenuse so it must be that c=2x+6.  It does not matter whether x is a or b because both parts are interchangeable. 

So the equation states, let a=x, b=2x+4 and c=2x+6.

Plug these conditions into the Pythagorean Theorem and solve.

a^2+b^2=c^2

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

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

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

      -4x^2 -24x -36   -4x^2 -24x -36

           x^2-8x-20=0

         (x+2)(x-10)=0

            x=-2, 10

x can not be negative, so x=10

Answer 2

First, draw a picture. I'm going to try to do this on here... we'll see how it goes.

 

x__________________________________Treasure

l

Castle Rock

 

So, we have a triangle with sides x, 2x+6, and 2x+4. From there, you can use the Pythagorean theorem to solve for x.

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

Solve it out, and then solve for x.