To solve for the distance between the top and bottom of the bridge using the given ratios and expressions for AC and EC, we can set up the equation based on the condition that the ratios are 1:1:
Given:
- AC = \(4x - 3\)
- EC = \(2x + 6\)
Since the triangles are in a 1:1 ratio, we can set the two expressions equal to each other:
\[ 4x - 3 = 2x + 6 \]
Now, we'll solve for \(x\):
- Subtract \(2x\) from both sides:
\[ 4x - 2x - 3 = 6 \]
This simplifies to:
\[ 2x - 3 = 6 \]
- Next, add 3 to both sides:
\[ 2x = 9 \]
- Now, divide both sides by 2:
\[ x = \frac{9}{2} = 4.5 \]
Now that we have the value of \(x\), we can substitute this back into either expression to find the lengths of AC and EC.
Let's find AC:
\[ AC = 4(4.5) - 3 = 18 - 3 = 15 \text{ ft} \]
Now let's find EC:
\[ EC = 2(4.5) + 6 = 9 + 6 = 15 \text{ ft} \]
Now we have both lengths AC and EC equal to 15 feet, confirming that the triangles are in the desired ratio of 1:1.
Therefore, the distance between the top and bottom of the bridge, based on the lengths we have calculated, is \(AC\) or \(EC\), both of which are:
\[ \boxed{15 \text{ ft}} \]
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