Question

1. open - ended if $overline{ab}$ has a mid - point of $m(-5,4)$, what are possible coordinates for points $a$ and $b$? explain your reasoning.

Explanation:

Step1: Recall mid - point formula

The mid - point formula between two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is $M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. Given $M(-5,4)$. Let $A=(x_a,y_a)$ and $B=(x_b,y_b)$. Then $\frac{x_a + x_b}{2}=-5$ and $\frac{y_a + y_b}{2}=4$. So $x_a + x_b=-10$ and $y_a + y_b = 8$.

Step2: Find possible points

There are infinitely many pairs of points. For example, if we let $x_a = 0$, then from $x_a + x_b=-10$, we have $x_b=-10$. If $y_a = 0$, then from $y_a + y_b = 8$, we have $y_b = 8$. So one possible pair is $A(0,0)$ and $B(-10,8)$. Another example, if $x_a=2$, then $x_b=-10 - 2=-12$, if $y_a = 2$ then $y_b=8 - 2 = 6$, so $A(2,2)$ and $B(-12,6)$ are also possible.

Answer:

There are infinitely many possible pairs of points $A$ and $B$. For example, $A(0,0)$ and $B(-10,8)$ or $A(2,2)$ and $B(-12,6)$.