To find the distance between the top and bottom of the bridge in feet, we need to solve for the variable [tex]\( x \)[/tex] and then use that value to determine the distance. The triangles are in a 1:1 ratio, meaning the sides [tex]\( AC \)[/tex] and [tex]\( EC \)[/tex] are equal.
Given:
[tex]\[ AC = 5x - 5 \][/tex]
[tex]\[ EC = 3x + 9 \][/tex]
Since [tex]\( AC = EC \)[/tex] due to the 1:1 ratio, we can set the two expressions equal to each other:
[tex]\[ 5x - 5 = 3x + 9 \][/tex]
To solve for [tex]\( x \)[/tex], we'll isolate the variable on one side of the equation:
1. Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ 5x - 3x - 5 = 3x - 3x + 9 \][/tex]
[tex]\[ 2x - 5 = 9 \][/tex]
2. Add 5 to both sides:
[tex]\[ 2x - 5 + 5 = 9 + 5 \][/tex]
[tex]\[ 2x = 14 \][/tex]
3. Divide both sides by 2:
[tex]\[ \frac{2x}{2} = \frac{14}{2} \][/tex]
[tex]\[ x = 7 \][/tex]
Now that we have the value of [tex]\( x = 7 \)[/tex], we can substitute [tex]\( x \)[/tex] back into either [tex]\( AC \)[/tex] or [tex]\( EC \)[/tex] to find the distance:
[tex]\[ AC = 5x - 5 \][/tex]
Substituting [tex]\( x = 7 \)[/tex]:
[tex]\[ AC = 5(7) - 5 = 35 - 5 = 30 \][/tex]
Alternatively, using [tex]\( EC \)[/tex]:
[tex]\[ EC = 3x + 9 \][/tex]
Substituting [tex]\( x = 7 \)[/tex]:
[tex]\[ EC = 3(7) + 9 = 21 + 9 = 30 \][/tex]
Both expressions confirm that the distance between the top and bottom of the bridge is [tex]\( \boxed{30} \)[/tex] feet.
