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We are given a diagram where \( \overline{CD} \) is the perpendicular bisector of \( \overline{AB} \), and we know the following:

- \( BD = 70 \, \text{feet} \)
- \( CD = 25 \, \text{feet} \)

Since \( CD \) is the perpendicular bisector of \( AB \), it means that \( C \) is the midpoint of \( AB \), and \( \triangle BCD \) is a right triangle with a right angle at \( C \).

To find the length of \( AB \), we can use the Pythagorean theorem on \( \triangle BCD \). The theorem states that:

\[
BD^2 = BC^2 + CD^2
\]

We are looking for the length of \( AB \), which is \( 2 \times BC \) because \( C \) is the midpoint of \( AB \). Let's first solve for \( BC \) using the Pythagorean theorem:

\[
BD^2 = BC^2 + CD^2
\]
\[
70^2 = BC^2 + 25^2
\]
\[
4900 = BC^2 + 625
\]
\[
BC^2 = 4900 - 625 = 4275
\]
\[
BC = \sqrt{4275}
\]

Now, I'll calculate the value of \( BC \).The length of \( BC \) is approximately 65.38 feet. Since \( AB \) is twice the length of \( BC \) (because \( C \) is the midpoint), the length of \( AB \) is:

\[
AB = 2 \times BC = 2 \times 65.38 = 130.77 \, \text{feet}
\]

Thus, the length of \( AB \) is approximately **130.77 feet**.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the significance of a perpendicular bisector in geometry?
2. How does the Pythagorean theorem apply to right triangles in construction problems?
3. How would the problem change if \( \overline{CD} \) was not a perpendicular bisector?
4. What other methods could be used to find the length of \( AB \) in a different context?
5. How would the trigonometric functions apply if the angles in the triangle were known?

**Tip**: Always double-check the geometric properties of a figure when solving word problems, especially whether lines are perpendicular or bisecting.

In the following diagram of one section of a bridge truss, CD is the perpendicular bisector of AB. If BD is 70 feet and CD is 25 feet, what is the length of AB?

Explore how to determine the length of segment AB in a bridge truss section, using the Pythagorean theorem. This problem utilizes the properties of perpendicular bisectors and right triangles, ideal for high school geometry students.

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