Question

1. the endpoints of cd are c(-2, -4) and d(6, 2). what is the y - coordinate of the midpoint of cd? type your answer in the box y =
2. the point (2, -3) is the midpoint of segment jk. point k is located at (-6,-3). what is the length of segment jk?

a. 2√7
b. 16
c. 8√2

Explanation:

Step1: Recall mid - point formula for y - coordinate

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $y_{mid}=\frac{y_1 + y_2}{2}$. For points $C(-2,-4)$ and $D(6,2)$, we have $y_1=-4$ and $y_2 = 2$.

Step2: Calculate the y - coordinate of the mid - point

$y=\frac{-4 + 2}{2}=\frac{-2}{2}=-1$

Step1: Recall mid - point formula for finding the other endpoint

Let the mid - point of segment $JK$ be $M(2,-3)$ and one endpoint $K(-6,-3)$. The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $x_{mid}=\frac{x_1 + x_2}{2}$ and $y_{mid}=\frac{y_1 + y_2}{2}$. We first find the $x$ - coordinate of $J$. Given $x_{mid}=2$, $x_2=-6$, then $2=\frac{x_1-6}{2}$, multiply both sides by 2: $4=x_1 - 6$, so $x_1=10$. Since $y_{mid}=-3$ and $y_2=-3$, then $-3=\frac{y_1-3}{2}$, multiply both sides by 2: $-6=y_1-3$, so $y_1=-3$. So the coordinates of $J$ are $(10,-3)$.

Step2: Use the distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For $J(10,-3)$ and $K(-6,-3)$, $x_1 = 10$, $y_1=-3$, $x_2=-6$, $y_2=-3$. Then $d=\sqrt{(-6 - 10)^2+(-3+3)^2}=\sqrt{(-16)^2+0^2}=\sqrt{256}=16$

Answer:

$-1$