Given that the midpoint of the line segment is [tex]\((11, -5)\)[/tex] and one endpoint of the line segment is [tex]\((-4, -8)\)[/tex], we are to find the coordinates of the other endpoint.
We use the midpoint formula, which is:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
where [tex]\( M \)[/tex] is the midpoint, and [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] are the coordinates of the two endpoints.
For our problem:
- The midpoint [tex]\(M\)[/tex] is [tex]\((11, -5)\)[/tex]
- One endpoint [tex]\((x_1, y_1)\)[/tex] is [tex]\((-4, -8)\)[/tex]
- We need to find the coordinates [tex]\((x_2, y_2)\)[/tex] of the other endpoint.
Let's set up the equations from the midpoint formula:
[tex]\[ 11 = \frac{-4 + x_2}{2} \][/tex]
[tex]\[ -5 = \frac{-8 + y_2}{2} \][/tex]
We solve these equations step-by-step.
Step 1: Solve for [tex]\(x_2\)[/tex]
[tex]\[ 11 = \frac{-4 + x_2}{2} \][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[ 22 = -4 + x_2 \][/tex]
Add 4 to both sides to solve for [tex]\(x_2\)[/tex]:
[tex]\[ x_2 = 22 + 4 \][/tex]
[tex]\[ x_2 = 26 \][/tex]
Step 2: Solve for [tex]\(y_2\)[/tex]
[tex]\[ -5 = \frac{-8 + y_2}{2} \][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[ -10 = -8 + y_2 \][/tex]
Add 8 to both sides to solve for [tex]\(y_2\)[/tex]:
[tex]\[ y_2 = -10 + 8 \][/tex]
[tex]\[ y_2 = -2 \][/tex]
Thus, the coordinates of the other endpoint are [tex]\((26, -2)\)[/tex].
The correct answer is [tex]\((26, -2)\)[/tex].
