Trigonometry Demystified, 2nd edition
Stan Gibilisco
Explanations for Quiz Answers in Chapter 2
1. In Fig. 2-10, we see no points in the upper-right quadrant, where both variables have positive values. That's the first quadrant. The correct choice is D. The point (-4,5) lies in the second quadrant; the point (-5,-3) lies in the third quadrant; the point (1,-6) lies in the fourth quadrant. The origin (0,0) doesn't lie in any quadrant.
2. To find the distance of the point (-4,5) from the origin (0,0), we must square both values, add the squares, and then take the positive square root of the result. If we call this distance r, then

r = [(-4)2 + 52]1/2
= (16 + 25)1/2
= 411/2
= 6.403

rounded off to three decimal places. The answer is B.
3. To find the distance r of the point (-5,-3) from the origin (0,0), we calculate

r = [(-5)2 + (-3)2]1/2
= (25 + 9)1/2
= 341/2
= 5.831

rounded off to three decimal places. The answer is D.
4. To find the distance r of the point (1,-6) from the origin (0,0), we calculate

r = [12 + (-6)2]1/2
= (1 + 36)1/2
= 371/2
= 6.083

rounded off to three decimal places. The answer is A.
5. To find the distance d between the points (-4,5) and (-5,-3) in Fig. 2-10, we must subtract the x values from each other, then subtract the y values from each other (in the same order as we did with the x values), then square each of those results separately, add the squares, and finally take the positive square root. We calculate

d = {[(-5) - (-4)]2 + (-3 - 5)2}1/2
= [(-5 + 4)2 + (-3 - 5)2]1/2
= [(-1)2 + (-8)2]1/2
= (1 + 64)1/2
= 651/2
= 8.062

rounded off to three decimal places. The correct choice is C.
6. To find the distance d between the points (-5,-3) and (1,-6), we calculate

d = {[1 - (-5)]2 + [(-6) - (-3)]2}1/2
= [(1 + 5)2 + (-6 + 3)2]1/2
= [62 + (-3)2]1/2
= (36 + 9)1/2
= 451/2
= 6.708

rounded off to three decimal places. The correct choice is C.
7. To find the distance d between the points (1,-6) and (-4,5), we calculate

d = {(-4 - 1)2 + [5 - (-6)]2}1/2
= [(-5)2 + (5 + 6)2]1/2
= [(-5)2 + 112]1/2
= (25 + 121)1/2
= 1461/2
= 12.083

rounded off to three decimal places. The correct choice is B.
8. To find the midpoint of a line segment in the Cartesian plane, we average the x values of the end-point coordinates to get the x value of the midpoint; then we average the y values of the end-point coordinates to get the y value of the midpoint. In Fig. 2-11, the end-point coordinates for line segment L are (-4,5) and (-5,-3). When we average the x values of the end points, we get

xm = [-4 + (-5)] / 2
= (-4 - 5) / 2
= -9/2

When we average the y values of the end points, we get

ym = [5 + (-3)] / 2
= (5 - 3) / 2
= 2/2
= 1

The coordinates of the midpoint are therefore

(xm,ym) = (-9/2,1)

The correct choice is C.
9. In Fig. 2-11, the end-point coordinates for line segment M are (-5,-3) and (1,-6). When we average the x values, we get

xm = (-5 + 1) / 2
= -4/2
= -2

When we average the y values, we get

ym = [-3 + (-6)] / 2
= (-3 - 6) / 2
= -9/2

The coordinates of the midpoint are therefore

(xm,ym) = (-2,-9/2)

The correct choice is A.
10. In Fig. 2-11, the end-point coordinates for line segment N are (1,-6) and (-4,5). When we average the x values, we get

xm = [1 + (-4)] / 2
= (1 - 4) / 2
= -3/2

When we average the y values, we get

ym = (-6 + 5) / 2
= -1/2

The coordinates of the midpoint are therefore

(xm,ym) = (-3/2,-1/2)

The correct choice is A.