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Lesson 152: Midpoint and Distance Formulas
In this lesson we'll learn some formulas that come up quite often in geometry. They are easy to understand, and easy to recreate if you happen to forget them.
Very often, we are asked to find the midpoint of the line segment that connects two points. All we really do is take the average of the x coordinates of the points, and the average of the y coordinates of the points. Those averages are the coordinates of the midpoint. The formula for the x coordinate of the midpoint is (x1 + x2) / 2, formula for the y coordinate of the midpoint is (y1 + y2) / 2.
Let's try an example. Find the midpoint of the line segment that connects the points (1, 4) and (5, 10). Taking the average of the x coordinates, we'll do (1+5) / 2 to get 3. Taking the average of the y coordinates, we'll do (4+10) / 2 to get 7. The midpoint is (3, 7). As always, if your coordinates involve negative numbers, you'll have to be very careful with your signed arithmetic. Note that sometimes the midpoint will be a fractional value such as 6.5. That is fine, just check that you did your computations correctly.
Sometimes we are asked to find the distance between two points.
Of course, if the points lie along a horizontal or vertical line, it
is quite easy to calculate the distance. It is not so obvious,
though, if they lie along a diagonal line. Here is the actual
formula for the distance between two points A and B:
This
formula looks rather scary, but it is actually just the Pythagorean
Theorem in disguise. Let's plot two points, A and B, and
connect them such that a right triangle is formed. See the
diagram a left.
The horizontal distance between the two points is just the difference in x coordinate values of the points. The vertical distance is just the difference in the y coordinate values. What we really just found are the lengths of the legs of the right triangle that we're working with. We know how to find the value of the hypotenuse using the Pythagorean Theorem. We add the squares of each leg, then take the square root of the sum. Look at the distance formula. That is all that it is. Let's practice with an example.
Find the distance between the points (2, 5) and (4, 9). The difference in x values is 2, and the difference in y values is 4. As always, be careful if negative numbers are involved. We have to square our values, and add them. We'll do 22 + 42 = 18. Now we must compute √18. We can simplify it as 3√2, which is how we will leave our answer.
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