Given the points $A(3, 2)$ and $B(-5, -3)$, what is the product of the coordinates of the midpoint of $\overline{AB}$? Express your answer as a common fraction.
I've tried to find the length of $\overline{AB}$ first by using Pythagorean Theorem, but I wasn't sure if it would work that way.
$8^2 + 5^2 = 64 + 25 = 89 = AB^2$, and so $AB = \sqrt{89}$. But then what would I do next to get the desired answer or is there another way?
The answer is $1/2$.
On
The midpoint of the given pair of points can be determined using the formula,
$\bar{x}=\frac{x_1+x_2}{2}$
$\bar{y}=\frac{y_1+y_2}{2}$
known as the Midpoint Formula.
So, for the given points A(3,2) and B(-5,-3) the midpoint of AB is easy to determined.
Applying the formula we have,
$\bar{x}=\frac{X_1+X_2}{2}$
$\bar{x}=\frac{3-5}{2}$
$\bar{x}={-1}$
We already have the first coordinate of the midpoint.
Now, to get the second coordinate of the midpoint
$\bar{y}=\frac{y_1+y_2}{2}$
$\bar{y}=\frac{2-3}{2}$
$\bar{y}=-\frac{1}{2}$
Therefore, the midpoint of AB is $\left(-1,-\frac{1}{2}\right)$.
The product of the coordinates is $(-1)\left(-\frac{1}{2}\right)=\frac{1}{2}$
To find the midpoints of two given points given their coordinates, you must find the average of the $x-$ and $y-$coordinates. For your points $A$ and $B,$ the midpoint has coordinates $$\left(\frac{3 - 5}{2}, \frac{2 - 3}{2}\right)$$ $$= \left(-1, -\frac{1}{2}\right).$$
The product of the coordinates is $(-1)\left(-\frac{1}{2}\right) = \boxed{\frac{1}{2}}.$