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(a) Since both points have the same ordinate (or y-value), d (b) Since both points have the same abscissa (or x-value), d (c) d (d) d 2(x2 2(x2 x1)2 x1)2 (y2 (y2 y1)2 y1)2 2(6 2[9 3)2 ( 3)]2 (8 x2 y2 4)2 ( 3 x1 y1 232 2)2 1 4
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2) and (3, 4); (c) (3, 4) and
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12.9 Applying the distance formula to a triangle (a) Find the lengths of the sides of a triangle whose vertices are A(1, 1), B(1, 4), and C(5, 1). (b) Show that the triangle whose vertices are G(2, 10), H(3, 2), and J(6, 4) is a right triangle.
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See Fig. 12-9.
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y G(2,10)
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B(1, 4) 3 2 1 A(1, 1) C(5, 1) x O 1 2 3 4 5 y (a) 4 3 2 1 O y (b) J(6,4) H(3,2) x 1 2 3 4 5 6
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Fig. 12-9
CHAPTER 12 Analytic Geometry
(a) AC
4 and AB
3; BC
( (2
3)2 3)
5. (10 2)2
(b) (GJ) (6 2) (4 10) 52; (HJ) (6 3) (4 2) 65. Since (GJ)2 (HJ)2 (GH)2, ^GHJ is a right triangle.
13; (GH)
12.10 Applying the distance formula to a parallelogram The coordinates of the vertices of a quadrilateral are A(2, 2), B(3, 5), C(6, 7), and D(5, 4). Show that ABCD is a parallelogram.
Solution
See Fig. 12-10, where we have AB CD BC AD 2(3 2(6 2(6 2(5 2)2 5)2 3)2 2)2 (5 (7 (7 (4 2)2 4)2 5)2 2)2 212 212 232 232 32 32 22 22 210 210 213 213
Thus, AB > CD and BC > AD. Since opposite sides are congruent, ABCD is a parallelogram.
y B(3, 5) 4 3 2 1 O y
C(6, 7)
D(5, 4) A(2, 2) x 1 2 3 4
Fig. 12-10
12.11 Applying the distance formula to a circle A circle is tangent to the x-axis and has its center at (6, 4). Where is the point (9, 7) with respect to the circle
Solution
Since the circle is tangent to the x-axis, AQ in Fig. 12-11 is a radius. By Principle 2, AQ By Principle 3, BQ 2(9 6)2 (7 4)2 greater than a radius so B is outside the circle. 232 32 4. 218. Since 218 is greater than 4, BQ is
y B(9, 7)
Q(6, 4)
x O y A(6, 0)
Fig. 12-11
CHAPTER 12 Analytic Geometry
12.4 Slope of a Line
PRINCIPLE 1:
If a line passes through the points P1 (x1, y1) and P2(x2, y2), then
4 Slope of P1P2
PRINCIPLE 2: PRINCIPLE 3:
y2 x2
y1 x1
The line whose equation is y
b has slope m.
The slope of a line equals the tangent of its inclination.
The inclination i of a line is the angle above the x-axis that is included between the line and the positive direction of the x-axis (see Fig. 12-12). In the figure,
4 Slope of P1P2 4 Slope of P1P2
y2 x2
y1 x1
tan i
The slope is independent of the order in which the end points are selected. Thus, y2 x2 y1 x1 y1 x1 y2 x2
4 slope of P2P1
Fig. 12-12
12.4A Positive and Negative Slopes
PRINCIPLE 4:
If a line slants upward from left to right, its inclination i is an acute angle and its slope is positive (Fig. 12-13).
y P y ( ) Q x (+) i x
y (+) x O y Slope of PQ = y + = =+ x + P i x (+) x x O
y Slope of PQ = y = = x +
Fig. 12-13
Fig. 12-14
CHAPTER 12 Analytic Geometry
If a line slants downward from left to right, its inclination is an obtuse angle and its slope is negative (Fig. 12-14). If a line is parallel to the x-axis, its inclination is 0 and its slope is 0 (Fig. 12-15). If a line is perpendicular to the x-axis, its inclination is 90 and it has no slope (Fig. 12-16).
PRINCIPLE 5:
PRINCIPLE 6: PRINCIPLE 7:
y P x O y y 0 Slope of PQ = = =0 x + x (+) y = 0 Q x x O
Q y P x = 0 x
y Slope of PQ = y + = (meaningless) x 0
Fig. 12-15
Fig. 12-16
12-4B Slopes of Parallel and Perpendicular Lines
PRINCIPLE 8:
Parallel lines have the same slope. tan i or m m , where m
In Fig. 12-17, l y l ; hence, corresponding angles i and i are equal, and tan i and m are the slopes of l and l .
PRINCIPLE 9:
Lines having the same slope are parallel to each other. (This is the converse of Principle 8.)
y l l x O
x O y
Fig. 12-17
Fig. 12-18
PRINCIPLE 10:
Perpendicular lines have slopes that are negative reciprocals of each other. (Negative reciprocals are numbers, such as 2 and 5, whose product is 1.) 5 2
1/m or mm 1, where m and m are the slopes of l and l .
Thus in Fig. 12-18, if l ' l , then m
PRINCIPLE 11:
Lines whose slopes are negative reciprocals of each other are perpendicular. (This is the converse of Principle 10.)
12.4C Collinear Points
Collinear points are points which lie on the same straight line. Thus, A, B, and C are collinear points here:
CHAPTER 12 Analytic Geometry
PRINCIPLE 12:
The slope of a straight line is constant all along the line.
Thus if PQ above is a straight line, the slope of the segment from A to B equals the slope of the segment from C to Q.
PRINCIPLE 13:
If the slope of a segment between a first point and a second equals the slope of the segment between either point and a third, then the points are collinear.
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