Alright, let's solve the problem step-by-step.
1. Understand the problem: We need to find the coordinates of endpoint [tex]\( G \)[/tex] given the midpoint [tex]\( M \)[/tex] and one endpoint [tex]\( H \)[/tex].
2. Midpoint formula: The midpoint [tex]\( M \)[/tex] of a segment with endpoints [tex]\( G(x_1, y_1) \)[/tex] and [tex]\( H(x_2, y_2) \)[/tex] is given by:
[tex]\[
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\][/tex]
3. Known values:
- [tex]\( M = (-6, -3) \)[/tex]
- [tex]\( H = (-4, 4) \)[/tex]
4. Set up the equations:
According to the midpoint formula:
[tex]\[
-6 = \frac{x_1 + (-4)}{2} \quad \text{and} \quad -3 = \frac{y_1 + 4}{2}
\][/tex]
5. Solve for [tex]\( x_1 \)[/tex] and [tex]\( y_1 \)[/tex]:
- For the x-coordinate:
[tex]\[
-6 = \frac{x_1 - 4}{2}
\][/tex]
Multiply both sides by 2:
[tex]\[
-12 = x_1 - 4
\][/tex]
Add 4 to both sides:
[tex]\[
x_1 = -12 + 4
\][/tex]
[tex]\[
x_1 = -8
\][/tex]
- For the y-coordinate:
[tex]\[
-3 = \frac{y_1 + 4}{2}
\][/tex]
Multiply both sides by 2:
[tex]\[
-6 = y_1 + 4
\][/tex]
Subtract 4 from both sides:
[tex]\[
y_1 = -6 - 4
\][/tex]
[tex]\[
y_1 = -10
\][/tex]
6. Conclusion: The coordinates of endpoint [tex]\( G \)[/tex] are [tex]\( (-8, -10) \)[/tex].
7. Check the options:
- Option a: [tex]\( (1, 3.5) \)[/tex]
- Option b: [tex]\( (-2, 11) \)[/tex]
- Option c: [tex]\( (-8, -10) \)[/tex]
- Option d: [tex]\( (-5, 0.5) \)[/tex]
The correct answer is:
c. (-8, -10)
