slope of AC = cl(b + a) and the slope of BD is cl(b - a). Since AC and BD are perpendicular,
But, AB = a and AD = Vfe
2 + c
2 = a = AB. So, A BCD is a rhombus.
Using coordinates, prove that a trapezoid with equal diagonals is isosceles.
As shown in Fig. 3-7, let the trapezoid ABCD. with parallel bases AS and CD, have A at the origin and
B on the positive .x-axis. Let D be (b, c) and C be (d, c), with b < d. By hypothesis, ~AC = ~BD,
Vc- + d~ = \J(b - a)
2 + c
2 , d
2 = (b-a)
2 , d=±(b-a). Case 1. d = b - a. Then b = d + a>d. contradicting b < d. Case 2. d=a-b. Then b = a-d, b
2 = (d-a)
2 .
Hence, AD = Vb
2 + c
2 =
V (rf ~ «)" + c
2 = BC. So, the trapezoid is isosceles.
Find the intersection of the lines x - 2y = 2 and 3* + 4y = 6.
We must solve x - 2y = 2 and 3x + 4y = 6 simultaneously. Multiply the first equation by 2, obtaining
2x - 4y = 4, and add this to the second equation. The result is 5x = 10, .v = 2. When x = 2, substitution in either equation yields y = 0. Hence, the intersection is the point (2, 0).
Find the intersection of the lines 4x + 5y = 10 and 5.v + 4y = 8.
Multiply the first equation by 5 and the second equation by 4, obtaining 20x + 25y = 50 and 20x + 16y =
32. Subtracting the second equation from the first, we get 9y = 18. y = 2. When y = 2, x = 0. So, the
intersection is (0, 2).
Find the intersection of the line y = 8x - 6 and the parabola y = 2x
2 .
Solve
y = 8x-6
and y = 2x
2
simultaneously. 2x
2 = 8x - 6, jt
2 = 4*--3,
*
2 -4* + 3 = 0,
(x -3)(x- 1) = 0, A-= 3 or x = l. When x = 3, y = 18, and when x=l. y = 2. Thus, the intersection consists of (3, 18) and (1,2).
Find the intersection of the line y = x - 3 and the hyperbola xy = 4.
We must solve y = x - 3 and xy = 4 simultaneously. Then x(x - 3) = 4, x
2 - 3x - 4 = 0,
(x-4)(x + l) = 0, x = 4 or x = -l.
When .v = 4, y = 1, and when .v=-l, y = -4. Hence, the
intersection consists of the points (4,1) and (-1, -4).
Let x represent the number of million pounds of chicken that farmers offer for sale per week, and let y represent
the number of dollars per pound that consumers are willing to pay for chicken. Assume that the supply equation
for chicken is y = 0.02* + 0.25, that is, 0.02* + 0.25 is the price per pound at which farmers are willing to
sell x million pounds. Assume also that the demand equation for chicken is y = -0.025* + 2.5, that is,
-0.025* + 2.5 is the price per pound at which consumers are willing to buy x million pounds per week. Find
the intersection of the graphs of the supply and demand equations.
Set 0.02* + 0.25 =-0.025* + 2.5, 0.045^ = 2.25, .v = 2.25/0.045 = 2250/45 = 50 million pounds. Then
y = 1.25 dollars per pound is the price.
Find the coordinates of the point on the line y - 2x + 1 that is equidistant from (0,0) and (5, -2).
Setting the distances from (x, y) to (0,0) and (5, -2) equal and squaring, we obtain x2 + y2 = (x - 5)2 +
(y + 2)
2 , *
2 +y
2 = jt
2 -Kb: + 25 + y
2 + 4>>+4, 10*-4y = 29. Substituting 2x + 1 for y in the last equation we obtain 10*-4(2* + l) = 29, 2* = 33, ,r=¥. Then y = 34. So, the desired point is (f, 34).
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CHAPTER 3
3.77
Fig. 3-7
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3.79
3.80
3.81
3.82
3.83
But, AB = a and AD = Vfe
2 + c
2 = a = AB. So, A BCD is a rhombus.
Using coordinates, prove that a trapezoid with equal diagonals is isosceles.
As shown in Fig. 3-7, let the trapezoid ABCD. with parallel bases AS and CD, have A at the origin and
B on the positive .x-axis. Let D be (b, c) and C be (d, c), with b < d. By hypothesis, ~AC = ~BD,
Vc- + d~ = \J(b - a)
2 + c
2 , d
2 = (b-a)
2 , d=±(b-a). Case 1. d = b - a. Then b = d + a>d. contradicting b < d. Case 2. d=a-b. Then b = a-d, b
2 = (d-a)
2 .
Hence, AD = Vb
2 + c
2 =
V (rf ~ «)" + c
2 = BC. So, the trapezoid is isosceles.
Find the intersection of the lines x - 2y = 2 and 3* + 4y = 6.
We must solve x - 2y = 2 and 3x + 4y = 6 simultaneously. Multiply the first equation by 2, obtaining
2x - 4y = 4, and add this to the second equation. The result is 5x = 10, .v = 2. When x = 2, substitution in either equation yields y = 0. Hence, the intersection is the point (2, 0).
Find the intersection of the lines 4x + 5y = 10 and 5.v + 4y = 8.
Multiply the first equation by 5 and the second equation by 4, obtaining 20x + 25y = 50 and 20x + 16y =
32. Subtracting the second equation from the first, we get 9y = 18. y = 2. When y = 2, x = 0. So, the
intersection is (0, 2).
Find the intersection of the line y = 8x - 6 and the parabola y = 2x
2 .
Solve
y = 8x-6
and y = 2x
2
simultaneously. 2x
2 = 8x - 6, jt
2 = 4*--3,
*
2 -4* + 3 = 0,
(x -3)(x- 1) = 0, A-= 3 or x = l. When x = 3, y = 18, and when x=l. y = 2. Thus, the intersection consists of (3, 18) and (1,2).
Find the intersection of the line y = x - 3 and the hyperbola xy = 4.
We must solve y = x - 3 and xy = 4 simultaneously. Then x(x - 3) = 4, x
2 - 3x - 4 = 0,
(x-4)(x + l) = 0, x = 4 or x = -l.
When .v = 4, y = 1, and when .v=-l, y = -4. Hence, the
intersection consists of the points (4,1) and (-1, -4).
Let x represent the number of million pounds of chicken that farmers offer for sale per week, and let y represent
the number of dollars per pound that consumers are willing to pay for chicken. Assume that the supply equation
for chicken is y = 0.02* + 0.25, that is, 0.02* + 0.25 is the price per pound at which farmers are willing to
sell x million pounds. Assume also that the demand equation for chicken is y = -0.025* + 2.5, that is,
-0.025* + 2.5 is the price per pound at which consumers are willing to buy x million pounds per week. Find
the intersection of the graphs of the supply and demand equations.
Set 0.02* + 0.25 =-0.025* + 2.5, 0.045^ = 2.25, .v = 2.25/0.045 = 2250/45 = 50 million pounds. Then
y = 1.25 dollars per pound is the price.
Find the coordinates of the point on the line y - 2x + 1 that is equidistant from (0,0) and (5, -2).
Setting the distances from (x, y) to (0,0) and (5, -2) equal and squaring, we obtain x2 + y2 = (x - 5)2 +
(y + 2)
2 , *
2 +y
2 = jt
2 -Kb: + 25 + y
2 + 4>>+4, 10*-4y = 29. Substituting 2x + 1 for y in the last equation we obtain 10*-4(2* + l) = 29, 2* = 33, ,r=¥. Then y = 34. So, the desired point is (f, 34).
18
CHAPTER 3
3.77
Fig. 3-7
3.78
3.79
3.80
3.81
3.82
3.83