Question

find the other endpoint of the line segment with the given endpoint and midpoint.

1. endpoint: (-5,4), midpoint: (-10,-6)
2. endpoint: (-8,8), midpoint: (5,-3)

-5 + -10 / 2 4 + -6 / 2
find the midpoint of each line segment.
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Explanation:

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Step1: Use mid - point formula for x - coordinate

The mid - point formula is $M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Let the given endpoint be $(x_1,y_1)=(-5,4)$ and the mid - point be $(M_x,M_y)=(-10,-6)$. For the x - coordinate, we have $M_x=\frac{x_1 + x_2}{2}$, so $-10=\frac{-5 + x_2}{2}$. Multiply both sides by 2: $-20=-5 + x_2$. Then add 5 to both sides: $x_2=-20 + 5=-15$.

Step2: Use mid - point formula for y - coordinate

For the y - coordinate, $M_y=\frac{y_1 + y_2}{2}$, so $-6=\frac{4 + y_2}{2}$. Multiply both sides by 2: $-12=4 + y_2$. Subtract 4 from both sides: $y_2=-12 - 4=-16$.

Step1: Use mid - point formula for x - coordinate

Let the given endpoint be $(x_1,y_1)=(-8,8)$ and the mid - point be $(M_x,M_y)=(5,-3)$. Using $M_x=\frac{x_1 + x_2}{2}$, we have $5=\frac{-8 + x_2}{2}$. Multiply both sides by 2: $10=-8 + x_2$. Add 8 to both sides: $x_2=10 + 8 = 18$.

Step2: Use mid - point formula for y - coordinate

Using $M_y=\frac{y_1 + y_2}{2}$, we have $-3=\frac{8 + y_2}{2}$. Multiply both sides by 2: $-6=8 + y_2$. Subtract 8 from both sides: $y_2=-6 - 8=-14$.

Step1: Identify endpoints

From the graph, the endpoints of the line segment are $(-1,-3)$ and $(3,-5)$.

Step2: Apply mid - point formula

The mid - point formula is $M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Here, $x_1=-1,x_2 = 3,y_1=-3,y_2=-5$. So, $M_x=\frac{-1+3}{2}=\frac{2}{2}=1$ and $M_y=\frac{-3+( - 5)}{2}=\frac{-3 - 5}{2}=\frac{-8}{2}=-4$.

Answer:

$(-15,-16)$

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