To find the missing endpoint of a line segment when given one endpoint and the midpoint, we can use the midpoint formula:

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

Here:

• The midpoint is $(2, 3)$,
• One endpoint is $(-8, -5)$,
• Let the other endpoint be $(x_2, y_2)$.

We can set up the system of equations for the midpoint:

$\frac{-8 + x_2}{2} = 2 \quad \text{and} \quad \frac{-5 + y_2}{2} = 3$

Let's solve for $x_2$ and $y_2$:

$\frac{-8 + x_2}{2} = 2 \implies -8 + x_2 = 4 \implies x_2 = 12$

$\frac{-5 + y_2}{2} = 3 \implies -5 + y_2 = 6 \implies y_2 = 11$

Thus, the missing endpoint is $(12, 11)$.

The correct answer is the option $(12, 11)$.

Would you like more details or have any questions? Here are 5 related questions to deepen your understanding:

1. How do you find the midpoint between two points?
2. Can you explain the properties of a line segment's midpoint?
3. How would you find the missing endpoint if the midpoint and both endpoints are in 3D?
4. What happens if the two endpoints are the same?
5. How does the midpoint formula relate to the average of the coordinates?

Tip: The midpoint formula is the average of the x-coordinates and y-coordinates of two endpoints.