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Again dividing each side by 2, we get (yp + yq)/2 = (yq + yp)/2 We ve shown that the coordinates in the ordered pair on the left-hand side of the original equation are equal to the corresponding coordinates in the ordered pair on the right-hand side The ordered pairs are identical, so the midpoint is the same in either direction
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To find a midpoint of a line segment in Cartesian two-space, you simply average the coordinates of the endpoints This method always works if the midpoint lies on a straight line segment between the two endpoints But you might wonder, How can we find the midpoint between two points along an arc connecting those points In a situation like that, we must determine the length of the arc Depending on the nature of the arc, that can be fairly hard, very hard, or almost impossible! Arc-length problems are beyond the scope of this book, but you ll learn how to solve them in Calculus Know-It-All
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Consider two points in the Cartesian plane, one of which is at the origin Show that the coordinate values of the midpoint are exactly half the corresponding coordinate values of the point not on the origin
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We can plug in (0,0) as the coordinates of either point in the general midpoint formula, and work things out from there First, let s suppose that point P is at the origin and the coordinates of point Q are (xq,yq) Then xp = 0 and yp = 0 If we call the coordinates of the midpoint (xm,ym), we have (xm,ym) = [(xp + xq)/2,(yp + yq)/2] = [(0 + xq)/2,(0 + yq)/2] = (xq /2,yq /2) Now, let Q be at the origin and let the coordinates of P be (xp,yp) In that case, we have (xm,ym) = [(xp + xq)/2,(yp + yq)/2] = [(xp + 0)/2,(yp + 0)/2] = (xp /2,yp /2)
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This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App A The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it!
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Practice Exercises
y 6 4 Origin = (0, 0) 2 x 6 4 2 2 ( 5, 3) 4 6 2 4 6
( 4, 5)
(1, 6)
Figure 1-10
Illustration for Problems 1 through 7
y 6 4 Origin = (0, 0) L 2 x 6 4 N ( 5, 3) 4 6 2 2 2 4 6
( 4, 5)
(1, 6)
Figure 1-11
Illustration for Problems 8 through 10
Cartesian Two-Space
1 What are the x and y coordinates of the points shown in Fig 1-10 2 Determine the distance of the point ( 4,5) from the origin in Fig 1-10 Using a calculator, round off the answer to three decimal places 3 Determine the distance of the point ( 5, 3) from the origin in Fig 1-10 Using a calculator, round it off to three decimal places 4 Determine the distance of the point (1, 6) from the origin in Fig 1-10 Using a calculator, round it off to three decimal places 5 Determine the distance between the points ( 4,5) and ( 5, 3) in Fig 1-10 Using a calculator, round it off to three decimal places 6 Determine the distance between the points ( 5, 3) and (1, 6) in Fig 1-10 Using a calculator, round it off to three decimal places 7 Determine the distance between the points (1, 6) and ( 4,5) in Fig 1-10 Using a calculator, round it off to three decimal places 8 Determine the coordinates of the midpoint of line segment L in Fig 1-11 Express the values in fractional and decimal form 9 Determine the coordinates of the midpoint of line segment M in Fig 1-11 Express the values in fractional and decimal form 10 Determine the coordinates of the midpoint of line segment N in Fig 1-11 Express the values in fractional and decimal form