Question

ABCD is a square. AM=11. What is the measure of BD?

Answer 1

In square ABCD, let the points be defined as follows:
- Point A is at (0, 0)
- Point B is at (s, 0)
- Point C is at (s, s)
- Point D is at (0, s)

The point M is the midpoint of side AB. Thus, M has coordinates:
\[ M = \left( \frac{s}{2}, 0 \right) \]

According to the problem, AM = 11. The length of segment AM can be calculated as:
\[
AM = \sqrt{\left( \frac{s}{2} - 0 \right)^2 + (0 - 0)^2} = \frac{s}{2}
\]
We set this equal to 11:
\[
\frac{s}{2} = 11
\]

Multiplying both sides by 2 gives:
\[
s = 22
\]

Now, we need to find the length of diagonal BD. The coordinates of points B and D are:
- Point B: (22, 0)
- Point D: (0, 22)

The length of diagonal BD can be calculated using the distance formula:
\[
BD = \sqrt{(x_D - x_B)^2 + (y_D - y_B)^2}
\]
Substituting for B and D, we get:
\[
BD = \sqrt{(0 - 22)^2 + (22 - 0)^2}
\]
\[
= \sqrt{(-22)^2 + (22)^2}
\]
\[
= \sqrt{484 + 484}
\]
\[
= \sqrt{968}
\]
\[
= \sqrt{4 \times 242} = 2\sqrt{242}
\]

We can also express \( \sqrt{242} \) as:
\[
\sqrt{242} = \sqrt{121 \times 2} = 11\sqrt{2}
\]

Consequently, the length of diagonal BD is:
\[
BD = 2 \cdot 11\sqrt{2} = 22\sqrt{2}
\]

Thus, the measure of BD is:
\[
\boxed{22\sqrt{2}}
\]