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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 $1/2$.

Martin Sleziak
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space
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2 Answers2

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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 $(-1)\left(-\frac{1}{2}\right) = \boxed{\frac{1}{2}}.$

K. Jiang
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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 $(-1)\left(-\frac{1}{2}\right)=\frac{1}{2}$

khey
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  • Thanks for sharing the formula - never knew that one and it is indeed good to know! It's just that in the last line it should be 1/2 as there are two negatives. – space Feb 23 '16 at 09:40
  • I hope that formula will help you a lot. I didn't notice my mistake last time. There i already edited it. – khey Feb 23 '16 at 16:53