Question

1. find the midpoint between the points (-4, -3) and (-1, 1). *

your answer

1. find the distance between the coordinates (4, 1) and (2, 5). *

your answer

1. find the distance between the coordinates (2, 3) and (-7, 0). *

9.48

1. the midpoint of line eg is at f(6, 8). if point e is located at (4, 0), where would point g be?

8, 16

Explanation:

Step1: Recall mid - point formula

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.
For points $(-4,-3)$ and $(-1,1)$, $x_1=-4,y_1 = - 3,x_2=-1,y_2 = 1$.

Step2: Calculate x - coordinate of mid - point

$x=\frac{-4+( - 1)}{2}=\frac{-4 - 1}{2}=\frac{-5}{2}=-2.5$

Step3: Calculate y - coordinate of mid - point

$y=\frac{-3 + 1}{2}=\frac{-2}{2}=-1$
The mid - point is $(-2.5,-1)$.

Step4: Recall 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 points $(4,1)$ and $(2,5)$, $x_1 = 4,y_1=1,x_2 = 2,y_2 = 5$.

Step5: Calculate distance

$d=\sqrt{(2 - 4)^2+(5 - 1)^2}=\sqrt{(-2)^2+4^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\approx4.47$

Step6: For the third distance problem

For points $(2,3)$ and $(-7,0)$, using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, where $x_1 = 2,y_1 = 3,x_2=-7,y_2 = 0$.
$d=\sqrt{(-7 - 2)^2+(0 - 3)^2}=\sqrt{(-9)^2+( - 3)^2}=\sqrt{81+9}=\sqrt{90}\approx9.49$ (close to the given 9.48).

Step7: Recall mid - point formula for finding an endpoint

Let the coordinates of $E=(x_1,y_1)=(4,0)$ and mid - point $F=(x_m,y_m)=(6,8)$. Let the coordinates of $G=(x_2,y_2)$.
The mid - point formula $x_m=\frac{x_1 + x_2}{2}$ and $y_m=\frac{y_1 + y_2}{2}$.
For the x - coordinate: $6=\frac{4 + x_2}{2}$, then $12=4 + x_2$, so $x_2=8$.
For the y - coordinate: $8=\frac{0 + y_2}{2}$, then $16=0 + y_2$, so $y_2 = 16$.

Answer:

1. $(-2.5,-1)$
2. $2\sqrt{5}\approx4.47$
3. $\sqrt{90}\approx9.49$
4. $(8,16)$