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ssrs barcode font free SOLVED PROBLEMS in ObjectiveC
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1; thus, tan i
12.13 Slopes of parallel or perpendicular lines 4 4 4 4 4 4 2 Find the slope of CD if (a) AB y CD and the slope of AB is 3; (b) AB ' CD and the slope of 4 AB is 3. 4 Solutions
(a) By Principle 8, slope of CD (b) By Principle 10, slope of CD
slope of AB 1 slope of AB
2 3. 1 3/4 4 . 3 12.14 Applying principles 9 and 11 to triangles and quadrilaterals Complete each of the following statements: (a) In quadrilateral ABCD, if the slopes of AB, BC, CD, and DA are 2, the quadrilateral is a ____ . (b) In triangle LMP, if the slopes of LM and MP are 5 and Solutions
(a) Since the slopes of the opposite sides are equal, ABCD is a parallelogram. In addition, the slopes of adjacent sides are negative reciprocals; hence, those sides are and ABCD is a rectangle. (b) Since the slopes of LM and MP are negative reciprocals, LM ' MP and the triangle is a right triangle. 2, 1, and 2
2, respectively, 1 5, then LMP is a ____ triangle.
12.15 Applying principle 12 (a) AB has a slope of 2 and points A, B, and C are collinear. What are the slopes of AC and BC (b) Find y if G(1, 4), H(3, 2), and J(9, y) are collinear. CHAPTER 12 Analytic Geometry
Solutions
(a) By Principle 12, AC and BC have a slope of 2. (b) By Principle 12, slope of GJ y 4.
slope of GH. Hence
y 4 4 , so that 1 8
1 and
12.5 Locus in Analytic Geometry
A locus of points is the set of points, and only those points, satisfying a given condition. In geometry, a line or curve (or set of lines or curves) on a graph is the locus of analytic points that satisfy the equation of the line or curve. Think of the locus as the path of a point moving according to a given condition or as the set of points satisfying a given condition. PRINCIPLE 1: The locus of points whose abscissa is a constant k is a line parallel to the yaxis; its equation is x k. (See Fig. 1219.) The locus of points whose ordinate is a constant k is a line parallel to the xaxis; its equation is y k. (See Fig. 1219.) PRINCIPLE 2: y y y=k x=k x O y y x x
= O y mx
Fig. 1219 Fig. 1220 PRINCIPLE 3: The locus of points whose ordinate equals the product of a constant m and its abscissa is a straight line passing through the origin; its equation is y mx. The constant m is the slope of the line. (See Fig. 1220.) PRINCIPLE 4: The locus of points whose ordinate and abscissa are related by either of the equations y mx b or y x y1 x1 m where m and b are constants, is a line (Fig. 1221). y In the equation y mx b, m is the slope and b is the yintercept. The equation x the line passes through the fixed point (x1, y1) and has a slope of m. y1 x1
m tells us that
x O y
y P(x, y) r x O y x x
y y1 =m x x1 (x1, y1) x
Fig. 1221 Fig. 1222 CHAPTER 12 Analytic Geometry
PRINCIPLE 5: The locus of points such that the sum of the squares of the coordinates is a constant is a circle whose center is the origin. x2 y2 r2 y2 r2. The constant is the square of the radius, and the equation of the circle is (see Fig. 1222). Note that for any point P(x, y) on the circle, x2 SOLVED PROBLEMS
12.16 Applying principles 1 and 2 Graph and give the equation of the locus of points (a) whose ordinate is the yaxis; (c) that are equidistant from the points (3, 0) and (5, 0). Solutions
(a) From Principle 2, the equation is y (b) From Principle 1, the equation is x (c) The equation is x

