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.