SAT Midpoint Formula

Correct Answer: D

Solution 1:

Since points $A,$ $B,$ and $C$ are collinear, they have the same $y-$ coordinate, -3. Therefore, $n=-3.$

We can find the length of $\overline{AC}$ by subtracting the $x-$ coordinates of points $A$ and $C$ .

$AC=3-(-7)=3+7=10.$

By definition, the midpoint $B$ lies halfway between $A$ and $C$ . Therefore, it is located 5 units to the right of point $A,$ and 5 units to the left of point $C.$ We add 5 to the $x-$ coordinate of point $A,$ or subtract 5 from the $x-$ coordinate of point $C$ to find that point $C$ is located at $(-2, -3)$ . So, $m=-2.$

We can now find $m-n$ , which is $-2-(-3)=-2+3=1.$

Solution 2:

Tip: Midpoint formula: $M=\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right).$
As in Solution 1, we find that since points $A,$ $B,$ and $C$ are collinear, they have the same $y-$ coordinate, -3, and that therefore $n=-3.$

Now that we know that $A=(-7, -3)$ and $C=(3, -3),$ we use the midpoint formula to find the coordinates of point $B.$

$\begin{aligned} B&=\left(\frac{-7+3}{2}, \frac{-3+(-3)}{2}\right)\\ B&=\left(\frac{-4}{2}, \frac{-6}{2}\right)\\ B&=(-2, -3)\\ \end{aligned}$

Therefore, $m=-2$ and $m-n=-2-(-3)=-2+3=1.$

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Incorrect Choices:

(A)
Tip: Read the entire question carefully.
Tip: When distributing, be careful with signs!
If you solve for $m+n,$ instead of for $m-n$ , you will get this wrong answer. Or, if you forget to distribute the negative sign when subtracting $n$ from $m$ , like this

$m-n=-2\ \fbox{-}(-3)=-2\ \fbox{-}3=\boxed{-5},$

you will get this wrong answer.

(B)
Tip: Read the entire question carefully.
If you solve for $n$ , instead of for $m-n$ , you will get this wrong answer.

(C)
Tip: Read the entire question carefully.
If you solve for $m$ , instead of for $m-n$ , you will get this wrong answer.

(E)
This wrong choice is just meant to confuse you.