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Question 1108310: solve for 
                                   ^
                                 / |\
                                /  | \
                              6/   |  \4
                              /    |   \
                             /     |    \
                            /___x__|_____\
                            |-----6-------|

Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52890) About Me  (Show Source):
You can put this solution on YOUR website!
.
From the left right-angled triangle you have this Pythagorean equation

x^2 + h^2 = 6^2.        (1)


From the right right-angled triangle you have this Pythagorean equation

(6-x)^2 + h^2 = 4^2.    (2)


Subtract eq(2) from  eq(1). You will get

x^2 - (6-x)^2 = 36 - 16,

x^2 - 36 + 12x - x^2 = 20,

12x = 20 + 36 = 56.


=======>  x = 56%2F12 = 14%2F3.

Solved.


Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


The triangle is isosceles, with two sides of length 6. Then using the side with length 4 as the base, the height of the triangle will be the length of the other leg of a right triangle with one leg 2 (half of the 4) and hypotenuse 6. So the height is
sqrt%286%5E2-2%5E2%29+=+sqrt%2836-4%29+=+sqrt%2832%29+=+4%2Asqrt%282%29

Then the area of the triangle is one-half base times height: 2%284%2Asqrt%282%29%29+=+8%2Asqrt%282%29

Then, using your diagram with 6 as the base, the area of the triangle is one-half base times the altitude you show (call it h):
8%2Asqrt%282%29+=+3h
h+=+8%2Asqrt%282%29%2F3

And then the unknown x we are looking for is the other leg of a right triangle with one leg h and hypotenuse 6:
x%5E2%2B%288%2Asqrt%282%29%2F3%29%5E2+=+6%5E2
x%5E2%2B128%2F9+=+36+=+324%2F9
x%5E2+=+196%2F9
x+=+14%2F3