Question

a new bridge structure requires triangles that are in a ratio of 1:1. if \\(ac = 5x - 5\\) and \\(ec = 3x + 9\\), find the distance between the top and bottom of the bridge, in feet.

Explanation:

Identify given values and geometric relationships

We are given that the two triangles, \(\triangle CAB\) and \(\triangle CED\), are in a ratio of \(1:1\). This means they are congruent.
The segment \(AE\) is a straight vertical line passing through \(C\), containing segments \(AC\) and \(EC\).
The total distance between the top and bottom of the bridge is the length of segment \(AE\), which is:

$$AE = AC + EC$$

We are given:

$$AC = 5x - 5$$

$$EC = 3x + 9$$

Set up the equation using congruence

Since the triangles are in a \(1:1\) ratio, their corresponding sides are equal.
Using the AA Similarity Postulate and Vertical Angles Theorem, we can confirm \(\triangle CAB \cong \triangle CED\).
Therefore, the corresponding vertical segments are equal:

$$AC = EC$$

Substitute the given algebraic expressions:

$$5x - 5 = 3x + 9$$

Solve for x

Subtract \(3x\) from both sides:

$$2x - 5 = 9$$

Add \(5\) to both sides:

$$2x = 14$$

Divide by \(2\):

$$x = 7$$

Calculate the lengths of AC and EC

Substitute \(x = 7\) back into the expressions:

$$AC = 5(7) - 5 = 35 - 5 = 30$$

$$EC = 3(7) + 9 = 21 + 9 = 30$$

Both segments are indeed equal to \(30\) feet.

Calculate the total distance AE

The total distance between the top and bottom of the bridge is:

$$AE = AC + EC = 30 + 30 = 60$$

Answer:

60