Finding the Midpoint

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We can find the midpoint between two points on a number line by calculating the arithmetic mean (or average value) of the numbers corresponding to the points In Cartesian xy coordinates, we must make two calculations First, we average the x values of the two points to get the x value of the point midway between Then, we average the y values of the points to get the y value of the point midway between

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A mini theorem Once again, imagine points P and Q in the Cartesian plane with the coordinates

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P = (xp,yp) and Q = (xq,yq) Suppose we want to find the coordinates of the midpoint That s the point that bisects a straight line segment connecting P and Q As before, we start out by choosing the point R below and

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Cartesian Two-Space

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y Point M lies midway between points P and Q Point P (xp, yp)

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Point M

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Point My x

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Point Q (xq, yq)

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Point Mx

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Point R (xp, yq) What are the coordinates of M

Figure 1-9

We can calculate the coordinates of the midpoint of a line segment whose endpoints are known

to the right that forms a right triangle PQR, as shown in Fig 1-9 Imagine a movable point M that we can slide freely along line segment PQ When we draw a perpendicular from M to side QR, we get a point Mx When we draw a perpendicular from M to side RP, we get a point My Consider the three right triangles MQMx, PMMy, and PQR The laws of basic geometry tell us that these triangles are similar, meaning that the lengths of their corresponding sides are in the same ratios According to the definition of similarity for triangles, we know the following two facts: Point Mx is midway between Q and R if and only if M is midway between P and Q Point My is midway between R and P if and only if M is midway between P and Q Now, instead of saying that M stands for movable point, let s say that M stands for midpoint In this case, the x value of Mx (the midpoint of line segment QR) must be the x value of M, and the y value of My (the midpoint of line segment RP) must be the y value of M

The general formula We ve reduced our Cartesian two-space midpoint problem to two separate number-line midpoint problems Side QR of triangle PQR is parallel to the x axis, and side RP of triangle PQR is parallel to the y axis We can find the x value of Mx by averaging the x values of Q and R When we do this and call the result xm, we get

xm = (xp + xq)/2

Finding the Midpoint

In the same way, we can calculate the y value of My by averaging the y values of R and P Calling the result ym, we have ym = (yp + yq)/2 We can use the mini theorem we finished a few moments ago to conclude that the coordinates of point M, the midpoint of line segment PQ, are (xm,ym) = [(xp + xq)/2,(yp + yq)/2]

An example Let s find the coordinates (xm,ym) of the midpoint M between the same two points for which we found the separation distance earlier in this chapter:

P = ( 5, 2) and Q = (7,3) When we plug xp = 5, yp = 2, xq = 7, and yq = 3 into the midpoint formula, we get (xm,ym) = [(xp + xq)/2,(yp + yq)/2] = [( 5 + 7)/2,( 2 + 3)/2] = (2/2,1/2) = (1,1/2)

Are you a skeptic

It seems reasonable to suppose the midpoint between points P and Q should not depend on whether we go from P to Q or from Q to P We can prove this by showing that for all real numbers xp, yp, xq, and yq, we have [(xp + xq)/2,(yp + yq)/2] = [(xq + xp)/2,(yq + yp)/2] This demonstration is easy, but let s go through it step-by-step to completely follow the logic For the x coordinates, the commutative law of addition tells us that xp + xq = xq + xp Dividing each side by 2 gives us (xp + xq)/2 = (xq + xp)/2 For the y coordinates, the commutative law says that yp + yq = yq + yp