To find the coordinates of point $D$, we can use the midpoint formula. The formula for the midpoint $M(x_m, y_m)$ of a line segment with endpoints $C(x_1, y_1)$ and $D(x_2, y_2)$ is:

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

Given:

• Midpoint $M = (-5, -5)$
• Point $C = (-1, -6)$

Let $D(x_2, y_2)$ represent the unknown point.

Using the midpoint formula:

$-5 = \frac{-1 + x_2}{2} \quad \text{(for the x-coordinates)}$ $-5 = \frac{-6 + y_2}{2} \quad \text{(for the y-coordinates)}$

Now, solve these equations:

1. For the x-coordinate: $-5 = \frac{-1 + x_2}{2}$ Multiply both sides by 2: $-10 = -1 + x_2$ Add 1 to both sides: $x_2 = -9$

2. For the y-coordinate: $-5 = \frac{-6 + y_2}{2}$ Multiply both sides by 2: $-10 = -6 + y_2$ Add 6 to both sides: $y_2 = -4$

Thus, the coordinates of point $D$ are $(-9, -4)$.

Would you like more details or have any questions about the solution?

Here are 5 related questions to expand this topic:

1. How would you find the distance between points $C$ and $D$?
2. What is the slope of the line passing through points $C$ and $D$?
3. Can you calculate the equation of the line passing through $C$ and $D$?
4. How would you solve for the coordinates of point $D$ if $C$ were different?
5. What happens to the midpoint if the coordinates of $C$ and $D$ are changed?

Tip: The midpoint formula is useful when working with geometry problems involving bisectors or symmetry!