Distance and midpoint
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Find the distance between the points A(1, 2) and B(4, 6).
Common mix-up
In this example, a student wrote d = 3 + 4 = 7.
The student added the coordinate differences without squaring anything, computing the distance as 3 + 4 = 7.
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See why this works
How far apart are A(1, 2) and B(4, 6)?
Two points give you a right triangle in disguise. The legs are the changes in x and y; the distance is the hypotenuse.
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2); midpoint = ((x1 + x2)/2, (y1 + y2)/2).
The distance between two points is the length of the hypotenuse of a right triangle you can sketch around them. For A(1, 2) and B(4, 6), the horizontal change is 4 - 1 = 3 and the vertical change is 6 - 2 = 4. Those two legs are the skeleton of the Pythagorean theorem.
Square the legs, add, and take the square root: sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5. The formula does exactly this with coordinates: sqrt((x2 - x1)^2 + (y2 - y1)^2). The most common slip is to subtract after squaring, or to skip the squares entirely and add 3 + 4, which gives 7 instead of 5.
The midpoint is a different animal: it is an average, not a length. To find it, average the x-coordinates and average the y-coordinates separately. For P(-1, 4) and Q(5, 2), the midpoint is ((-1 + 5)/2, (4 + 2)/2) = (2, 3). The midpoint of a segment always lies exactly halfway along it.
Distance and midpoint are two formulas with two different operations: distance builds a right triangle and uses squares, while midpoint splits each coordinate in half. When a problem gives two points, decide first whether it wants a length or a location, and the formula follows.
Keep this idea
Distance needs differences and squares; midpoint needs averages. Either way, handle the x and y coordinates separately and label the answer.
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