A 125-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 25 ft longer that the height that it reaches on the side of the bridge. Find the horizontal and vertical distances spanned by this brace.

Consider a right triangle formed by the horizontal distance H, the vertical distance V, and the diagonal brace, L=125'.

Since
H=V+25
apply Pythagoras theorem:
H²+V²=L²
(V+25)²+V²=125²
Expand and simplify:
2V²+50V+625 = 15625
or
2V²+50V-15000=0

Solve for V by factoring or quadratic formula. Reject negative value for x.

Post your answer for checking, or back-substitute to check.

To find the horizontal and vertical distances spanned by the brace, let's denote the horizontal distance as x and the vertical distance as y.

From the given information, we can establish two equations:

1. The Pythagorean theorem:
x^2 + y^2 = 125^2

2. The relationship between the horizontal and vertical distances:
x = y + 25

We can solve this system of equations to find the values of x and y.

First, substitute the second equation into the first equation:
(y + 25)^2 + y^2 = 125^2

Expand and simplify the equation:
y^2 + 50y + 625 + y^2 = 15625

Combine like terms:
2y^2 + 50y + 625 = 15625

Rearrange the equation:
2y^2 + 50y - 15000 = 0

Divide the equation by 2 to simplify:
y^2 + 25y - 7500 = 0

Now we have a quadratic equation. We can solve it by factoring or using the quadratic formula. Let's use the quadratic formula:

y = (-b ± √(b^2 - 4ac)) / (2a)

where a = 1, b = 25, and c = -7500:

y = (-25 ± √(25^2 - 4*1*(-7500))) / (2*1)

Simplifying the square root:
y = (-25 ± √(625 + 30000)) / 2

y = (-25 ± √(30625)) / 2

y = (-25 ± 175) / 2

So, we have two possible values for y:
1. y = (-25 + 175) / 2 = 75
2. y = (-25 - 175) / 2 = -100

Since the height cannot be negative in this case, we can discard the second solution, leaving us with y = 75.

Substituting the value of y back into the equation x = y + 25:
x = 75 + 25 = 100

Therefore, the horizontal distance spanned by the brace is 100 ft, and the vertical distance spanned is 75 ft.