Question

name:____ date:____
midpoint
$\left(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2}\
ight)$
to find the point in the middle of two other points, we ____ the x coordinates then __ by 2. we __ the y coordinates then ____ by 2.
distance
$(x_1,y_1),(x_2,y_2)$
formula:
weighted average
find $ab$ when $a(-4,1)$ and $b(3,-1)$
find the midpoint of $g(7,-5)$ and $h(9,-1)$
the coordinate 2 has a weight of 2, the coordinate 5 has a weight of 1, and the coordinate 7 has a weight of 3.
find the distance of $pq$ if $p(-8,3)$ and $q(-2,7)$
find the length of $mx$
find the midpoint of this segment.

Explanation:

Step1: Mid - point concept

To find the mid - point of two points, for the x - coordinates, we add them and divide by 2. For the y - coordinates, we add them and divide by 2.

Step2: Distance formula

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

Step3: Find distance between A and B

Let $A(-4,1)$ and $B(3,-1)$.
$x_1=-4,y_1 = 1,x_2=3,y_2=-1$
$d=\sqrt{(3-(-4))^2+(-1 - 1)^2}=\sqrt{(3 + 4)^2+(-2)^2}=\sqrt{49 + 4}=\sqrt{53}$.

Step4: Find mid - point of G and H

Let $G(7,-5)$ and $H(9,-1)$.
$x_1 = 7,y_1=-5,x_2=9,y_2=-1$
Mid - point $(\frac{7 + 9}{2},\frac{-5+( - 1)}{2})=(8,-3)$.

Step5: Weighted average

The weighted average formula is $\bar{x}=\frac{\sum_{i = 1}^{n}w_ix_i}{\sum_{i=1}^{n}w_i}$.
Here, $x_1 = 2,w_1=2,x_2=5,w_2 = 1,x_3=7,w_3=3$
$\bar{x}=\frac{2\times2+1\times5+3\times7}{2 + 1+3}=\frac{4 + 5+21}{6}=\frac{30}{6}=5$.

Step6: Find distance between P and Q

Let $P(-8,3)$ and $Q(-2,7)$.
$x_1=-8,y_1 = 3,x_2=-2,y_2=7$
$d=\sqrt{(-2-(-8))^2+(7 - 3)^2}=\sqrt{(-2 + 8)^2+4^2}=\sqrt{36 + 16}=\sqrt{52}=2\sqrt{13}$.

Step7: Find length of MX

If $XM=5x-8$ and $MY = 9x-12$ and M is the mid - point of XY, then $5x-8=9x-12$.
$9x-5x=12 - 8$
$4x=4$
$x = 1$
Length of $MX=5\times1-8=-3$ (distance can't be negative, there may be a wrong - sign in the problem setup, assuming $XM$ and $MY$ are non - negative, we should have $MX = 8 - 5x$ and $MY=9x - 12$, then $8-5x=9x-12$, $14x=20$, $x=\frac{10}{7}$, $MX=8-5\times\frac{10}{7}=8-\frac{50}{7}=\frac{56 - 50}{7}=\frac{6}{7}$).

Step8: Mid - point of a segment (general)

For a segment with endpoints $(x_1,y_1)$ and $(x_2,y_2)$, the mid - point is $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$.

Answer:

Mid - point: Add, divide, add, divide.
Distance formula: $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$
Distance $AB=\sqrt{53}$
Mid - point of $GH=(8,-3)$
Weighted average = 5
Distance $PQ=2\sqrt{13}$
Length of $MX=\frac{6}{7}$ (after correction)
Mid - point of segment: $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$