Example SECTION XAXIS  the horizontal number line. YAXIS  the vertical number line ORIGIN  the point where the xaxis and yaxis cross


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1 CHAPTER 13 SECTION 131 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants XAXIS  the horizontal number line YAXIS  the vertical number line ORIGIN  the point where the xaxis and yaxis cross 131 The Distance Formula D = [(x 2 x 1 ) 2 + (y 2 y 1 ) 2 ] ½ Example Find the distance between points A(4, 2) and B(7, 2) d = 5 1
2 132 Theorem An equation of the circle with center (a,b) and radius r is r 2 = (x a) 2 + (yb) 2 Example Find an equation of the circle with center (2,5) and radius 3. (x + 2) 2 + (y 5) 2 = 9 Example Find the center and the radius of the circle with equation (x1) 2 + (y2) 2 = 9. (1, 2), r = 3 SECTION 132 Slope of a Line FORMULA FOR SLOPE m = change in ycoordinate change in xcoordinate Or m = rise run SLOPE is the ratio of vertical change to the horizontal change. The variable m is used to represent slope. 2
3 SLOPE OF A LINE m = y 2 y 1 x 2 x 1 HORIZONTAL LINE a horizontal line containing the point (a, b) is described by the equation y = b and has slope of 0 VERTICAL LINE a vertical line containing the point (c, d) is described by the equation x = c and has no slope Slopes Lines with positive slope rise to the right. Lines with negative slope fall to the right. The greater the absolute value of a line s slope, the steeper the line Theorem 133 SECTION 133 Parallel and Perpendicular Lines Two nonvertical lines are parallel if and only if their slopes are equal 3
4 Theorem 134 Two nonvertical lines are perpendicular if and only if the product of their slopes is  1 parallel to the line containing points M and N. M(2, 5) and N(0, 1) parallel to the line containing points M and N. M(3, 5) and N(0, 6) parallel to the line containing points M and N. M(2, 6) and N(2, 1) perpendicular to the line containing points M and N. M(4, 1) and N(5, 2) perpendicular to the line containing points M and N. M(3, 5) and N(0, 6) 4
5 perpendicular to the line containing points M and N. M(2, 6) and N(2, 1) Determine whether each pair of lines is parallel, perpendicular, or neither 7x + 2y = 14 7y = 2x  5 Determine whether each pair of lines is parallel, perpendicular, or neither 5x + 3y = 2 3x 5y = 15 Determine whether each pair of lines is parallel, perpendicular, or neither 2x 3y = 6 8x 4y = 4 DEFINITIONS Vectors SECTION 134 Vector any quantity such as force, velocity, or acceleration, that has both size (magnitude) and direction 5
6 Vector Vector AB is read vector AB and is equal to the ordered pair (change in x, change in y) DEFINITIONS Magnitude of a vector is the length of the arrow from point A to point B and is denoted by the symbol AB Scalar Multiple Use the Pythagorean Theorem or the Distance Formula to find the magnitude of a vector. In general, if the vector PQ = (a,b) then kpq = (ka, kb) Equivalent Vectors Vectors having the same magnitude and the same direction. Perpendicular Vectors Two vectors are perpendicular if the arrows representing them have perpendicular directions. 6
7 Parallel Vectors Two vectors are parallel if the arrows representing them have the same direction or opposite directions. Adding Vectors (a,b) + (c,d) = (a+c, b+d) EXAMPLE Given: Points P(5,4) and Q(1,2) EXAMPLE Determine whether (6,3) and (4,2) are parallel or perpendicular. Find PQ Find PQ EXAMPLE Determine whether (6,3) and (2,4) are parallel or perpendicular. SECTION 135 The Midpoint Formula 7
8 Midpoint Formula M( x 1 + x 2, y 1 + y 2 ) 2 2 Example Find the midpoint of the segment joining the points (4, 6) and (3, 2) M(1/2, 2) SECTION 136 LINEAR EQUATION is an equation whose graph is a straight line. Graphing Linear Equations The graph of any equation that can be written in the form Ax + By = C 136 Standard Form Where A and B are not both zero, is a line THEOREM The slope of the line Ax + By = C (B 0) is  A/B and Yintecept = C/B 8
9 Theorem 137 Slope Intercept form y = mx + b where m is the slope and b is the y intercept Write an equation of a line with the given y intercept and slope m=3 b = 6 Theorem 138 PointSlope Form Writing Linear Equations SECTION 137 An equation of the line that passes through the point (x 1, y 1 ) and has slope m is y y 1 = m (x x 1 ) Write an equation of a line with the given slope and through a given point m=2 P(1, 3) END 9
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