Question

1. given the endpoint (5,2) and the mid - point (10, - 2). find the other endpoint.
2. find the distance between: (5.7, - 4.1) & (8, 5.3)
3. find the midpoint of: (4,7) & (-2,1)
4. find the distance between: (0, - 1) & (2,0)
5. find the midpoint of: (-4,4) & (-2,2)
6. find the distance between: (-3, - 1) & (1, - 1)

Explanation:

8.

Step1: Recall mid - point formula

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Let $(x_1,y_1)=(5,2)$ and $M=(10,-2)$. We need to find $(x_2,y_2)$.
For the x - coordinate: $\frac{5 + x_2}{2}=10$. Multiply both sides by 2: $5+x_2 = 20$. Then subtract 5 from both sides: $x_2=20 - 5=15$.
For the y - coordinate: $\frac{2 + y_2}{2}=-2$. Multiply both sides by 2: $2+y_2=-4$. Then subtract 2 from both sides: $y_2=-4 - 2=-6$.

Step1: Recall distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Here, $(x_1,y_1)=(5.7,-4.1)$ and $(x_2,y_2)=(8,5.3)$.

Step2: Calculate differences

$x_2 - x_1=8 - 5.7 = 2.3$ and $y_2 - y_1=5.3-(-4.1)=5.3 + 4.1 = 9.4$.

Step3: Calculate square of differences

$(x_2 - x_1)^2=(2.3)^2 = 5.29$ and $(y_2 - y_1)^2=(9.4)^2=88.36$.

Step4: Sum of squares and square - root

$d=\sqrt{5.29+88.36}=\sqrt{93.65}\approx9.68$

Step1: Recall mid - point formula

The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Given $(x_1,y_1)=(4,7)$ and $(x_2,y_2)=(-2,1)$.

Step2: Calculate x - coordinate of mid - point

$\frac{4+( - 2)}{2}=\frac{4 - 2}{2}=1$.

Step3: Calculate y - coordinate of mid - point

$\frac{7 + 1}{2}=\frac{8}{2}=4$.

Answer:

$(15,-6)$

9.