Question

name: date: period: midpoint (x1 + x2 / 2, y1 + y2 / 2) 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 (x1,y1),(x2,y2) 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 rule for x - coordinates

Add the x - coordinates, then divide by 2.

Step2: Mid - point rule for y - coordinates

Add the y - coordinates, then divide by 2.

Step3: Distance formula

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

Step4: Weighted average formula

The weighted average $\bar{x}=\frac{\sum_{i = 1}^{n}w_ix_i}{\sum_{i = 1}^{n}w_i}$, where $w_i$ is the weight of $x_i$.

Step5: Find the distance between $A(-4,1)$ and $B(3,-1)$

$d_{AB}=\sqrt{(3-(-4))^2+((-1)-1)^2}=\sqrt{(3 + 4)^2+(-2)^2}=\sqrt{49 + 4}=\sqrt{53}$.

Step6: Find the mid - point of $G(7,-5)$ and $H(9,-1)$

The mid - point $M=(\frac{7 + 9}{2},\frac{-5+( - 1)}{2})=(8,-3)$.

Step7: Calculate the weighted average

The weighted average $\bar{x}=\frac{2\times2+1\times5+3\times7}{2 + 1+3}=\frac{4 + 5+21}{6}=\frac{30}{6}=5$.

Step8: Find the distance between $P(-8,3)$ and $Q(-2,7)$

$d_{PQ}=\sqrt{((-2)-(-8))^2+(7 - 3)^2}=\sqrt{(-2 + 8)^2+4^2}=\sqrt{36+16}=\sqrt{52}=2\sqrt{13}$.

Step9: Solve for $x$ in $5x+8=9x - 12$

$9x-5x=8 + 12$, $4x=20$, $x = 5$. Then the length of $MX=5x+8=5\times5+8=33$.

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 $MX = 33$