Question

the midpoint of $overline{gh}$ is $(1, - 4)$. the coordinates of point $g$ are $(3, - 5)$. what are the coordinates of point $h$? a. $(1,0)$ b. $(-1,-3)$ c. $(2,-\frac{9}{2})$ d. $(5,-6)$ e. $(5,-13)$

Explanation:

Step1: Recall mid - point formula

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Let the coordinates of point $G=(x_1,y_1)=(3,-5)$ and the coordinates of point $H=(x_2,y_2)$. The mid - point $M=(1,-4)$.

Step2: Solve for $x_2$

We know that $\frac{x_1 + x_2}{2}=1$. Substitute $x_1 = 3$ into the equation: $\frac{3 + x_2}{2}=1$. Multiply both sides by 2: $3+x_2 = 2$. Then subtract 3 from both sides: $x_2=2 - 3=-1$.

Step3: Solve for $y_2$

We know that $\frac{y_1 + y_2}{2}=-4$. Substitute $y_1=-5$ into the equation: $\frac{-5 + y_2}{2}=-4$. Multiply both sides by 2: $-5 + y_2=-8$. Then add 5 to both sides: $y_2=-8 + 5=-3$.

Answer:

B. $(-1,-3)$