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A 5-ft diagonal brace on a bridge connects a support of the center of the bridge to a side support on the bridge. The horizontal distance that it spans is 1 ft longer than the height that it reaches on the side of the bridge. Find the horizontal and vertical distances spanned by this brace.
Let h = the height reached by the support
Then the horizontal distance = h + 1
This forms a right triangle with hypotenuse = 5:
Using the Pythagorean theorem we can write
h^2 + (h+1)^2 = 5^2
Solve for h:
h^2 + h^2 + 2h + 1 = 25
2h^2 + 2h - 24 = 0
This can be factored as
2(h+4)(h-3) = 0
Taking the positive solution, we have h=3
So the vertical distance is 3, and the horizontal distance is 4.
Note that this is a 3-4-5 right triangle