Question

what is the mid - point for a segment with the following endpoints? (4,5) and (10,3)
(\\(\frac{x_1 + x_2}{2}\\),\\(\frac{y_1 + y_2}{2}\\))
midpoint

1. what is the mid - point for a segment with the following endpoints? (-2,-8) and (-11,20)

(\\(\frac{x_1 + x_2}{2}\\),\\(\frac{y_1 + y_2}{2}\\))
midpoint

1. calculate the length between the following points using the distance formula.

(1,6) and (7,14)
\\(\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\\)
length

1. calculate the length between the following points using the distance formula.

(-3,-2) and (9,3)
\\(\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\\)
length
optional: complete problems 10, 11, 13, 15, 17 - 20, 24, 29 from your textbook on separate lined paper. when you are finished, check the odd problems for solutions at the back of the textbook.

Explanation:

Step 1: Find mid - point formula application for (4,5) and (10,3)

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Here $x_1 = 4$, $x_2=10$, $y_1 = 5$, $y_2 = 3$.
$(\frac{4+10}{2},\frac{5 + 3}{2})=(\frac{14}{2},\frac{8}{2})=(7,4)$

Step 2: Find mid - point formula application for (-2,-8) and (-11,20)

Using the mid - point formula with $x_1=-2$, $x_2=-11$, $y_1=-8$, $y_2 = 20$.
$(\frac{-2+( - 11)}{2},\frac{-8 + 20}{2})=(\frac{-13}{2},\frac{12}{2})=(-6.5,6)$

Step 3: Find distance formula application for (1,6) and (7,14)

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 $x_1 = 1$, $x_2=7$, $y_1 = 6$, $y_2 = 14$.
$d=\sqrt{(7 - 1)^2+(14 - 6)^2}=\sqrt{6^2+8^2}=\sqrt{36 + 64}=\sqrt{100}=10$

Step 4: Find distance formula application for (-3,-2) and (9,3)

Using the distance formula with $x_1=-3$, $x_2=9$, $y_1=-2$, $y_2 = 3$.
$d=\sqrt{(9-( - 3))^2+(3-( - 2))^2}=\sqrt{(9 + 3)^2+(3 + 2)^2}=\sqrt{12^2+5^2}=\sqrt{144+25}=\sqrt{169}=13$

Answer:

1. Mid - point: (7,4)
2. Mid - point: (-6.5,6)
3. Length: 10
4. Length: 13