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.