Problema Solution
A 10-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 2 ft longer than the height that it reaches on the side of the bridge. Find the horizontal and vertical distances spanned by this brace.
Answer provided by our tutors
Note that the horizontal distance that it spans, the vertical distance that it spans, and the length of the brace form a right triangle.
Let x denote the vertical distance that it spans. Then, since it spans 1 more foot horizontally, it spans x + 2 feet horizontally.
By the Pythagorean Theorem:
x^2 + (x + 2)^2 = 10^2
==> x^2 + (x^2 + 4x + 4) = 100, by expanding
==> 2x^2 + 4x + 4 = 100
==> 2x^2 + 4x - 96 = 0, by setting the right side equal to zero
==> 2(x + 8)(x - 6) = 0, by factoring
==> x = -8 and x = 6, by the zero-product property.
Since lengths cannot be negative, x = 6.
Therefore, the brace spans 6 feet vertically and 6 + 2 = 8 feet horizontally.