Math / Geometry

QuestionC A B A new bridge structure requires triangles that are in a ratio of 1:1. If AC = 4x-3 and EC = 2x + 6, find the distance between the top and bottom of the bridge, in feet. O 4.5 ft 15 ft 18 ft O 30 ft

Studdy Solution

STEP 1

1. The triangles in question are similar triangles with a ratio of 1:1.
2. The segments AC and EC are parts of these triangles.
3. We need to find the value of $x$ such that the segments are equal due to the 1:1 ratio.
4. Once $x$ is found, we can calculate the distance between the top and bottom of the bridge.

STEP 2

1. Set up the equation based on the given ratio.
2. Solve for $x$ .
3. Calculate the distance using the value of $x$ .

STEP 3

Since the triangles are in a ratio of 1:1, the lengths of corresponding sides are equal. Therefore, we set the expressions for AC and EC equal to each other:
$4x - 3 = 2x + 6$

STEP 4

Solve the equation for $x$ .
First, subtract $2x$ from both sides:
$4x - 2x - 3 = 6$ $2x - 3 = 6$
Next, add 3 to both sides:
$2x = 6 + 3$ $2x = 9$
Finally, divide both sides by 2:
$x = \frac{9}{2}$ $x = 4.5$

STEP 5

Substitute the value of $x$ back into either expression to find the distance. Using AC:
$AC = 4x - 3$ $AC = 4(4.5) - 3$ $AC = 18 - 3$ $AC = 15$
Therefore, the distance between the top and bottom of the bridge is 15 feet.
The distance between the top and bottom of the bridge is:
$\boxed{15 \text{ ft}}$

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