Question

warm - up: use the coordinate grid to help you answer the following questions: 1. what is the midpoint of the segment connecting (1,2) and (5,2)? 2. what is the midpoint of the segment connecting (5,2) and (5,10)? 3. what is the midpoint of the segment connecting (1,2) and (5,10)? 4. how does the word average relate to finding midpoints?

Explanation:

Step1: Recall mid - point formula

The mid - point formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$.

Step2: Solve for question 1

For the points $(1,2)$ and $(5,2)$, $x_1 = 1,x_2 = 5,y_1 = 2,y_2 = 2$.
$x$ - coordinate of mid - point: $\frac{1 + 5}{2}=\frac{6}{2}=3$
$y$ - coordinate of mid - point: $\frac{2+2}{2}=\frac{4}{2}=2$
The mid - point is $(3,2)$.

Step3: Solve for question 2

For the points $(5,2)$ and $(5,10)$, $x_1 = 5,x_2 = 5,y_1 = 2,y_2 = 10$.
$x$ - coordinate of mid - point: $\frac{5 + 5}{2}=\frac{10}{2}=5$
$y$ - coordinate of mid - point: $\frac{2 + 10}{2}=\frac{12}{2}=6$
The mid - point is $(5,6)$.

Step4: Solve for question 3

Same as question 2, for the points $(1,2)$ and $(5,10)$, $x_1 = 1,x_2 = 5,y_1 = 2,y_2 = 10$.
$x$ - coordinate of mid - point: $\frac{1+5}{2}=3$
$y$ - coordinate of mid - point: $\frac{2 + 10}{2}=6$
The mid - point is $(3,6)$.

Step5: Explain the relation of "average" to mid - points

The mid - point of two points in a coordinate plane is found by taking the average of the $x$ - coordinates and the average of the $y$ - coordinates of the two endpoints. For example, if we have two points $(x_1,y_1)$ and $(x_2,y_2)$, the $x$ - coordinate of the mid - point $M_x=\frac{x_1 + x_2}{2}$ which is the average of the $x$ values, and the $y$ - coordinate of the mid - point $M_y=\frac{y_1 + y_2}{2}$ which is the average of the $y$ values.

Answer:

1. $(3,2)$
2. $(5,6)$
3. $(3,6)$
4. The mid - point of two points is found by taking the average of the $x$ - coordinates and the average of the $y$ - coordinates of the endpoints.