We are given the following:

• Points $A(-1,1)$, $B(-4,5)$, and $C(1,6)$.
• $M$ is the midpoint of segment $\overline{AC}$.

To find $BM$, we need to:

Step 1: Find the coordinates of point $M$

Since $M$ is the midpoint of $\overline{AC}$, we can calculate the midpoint coordinates using the midpoint formula:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Substituting the coordinates of $A(-1, 1)$ and $C(1, 6)$:

$M = \left( \frac{-1 + 1}{2}, \frac{1 + 6}{2} \right) = (0, 3.5)$

So, $M(0, 3.5)$.

Step 2: Use the distance formula to find $BM$

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Substituting the coordinates of $B(-4, 5)$ and $M(0, 3.5)$:

$BM = \sqrt{(0 - (-4))^2 + (3.5 - 5)^2}$ $BM = \sqrt{(4)^2 + (-1.5)^2} = \sqrt{16 + 2.25} = \sqrt{18.25}$

Thus, $BM \approx 4.27$ units.

Would you like more details on any part of the solution?

Here are some related questions:

1. How do we calculate the distance between two points in 3D space?
2. How can we use the Pythagorean theorem to verify the distance formula?
3. What are the coordinates of the centroid of triangle ABC?
4. How do you find the equation of the line passing through two points?
5. How would we find the area of triangle ABC using its vertices?

Tip: When working with midpoints, always double-check that you've averaged both the x- and y-coordinates separately for accuracy!