Example SECTION X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

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1 CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal number line Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross 13-1 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 13-2 Theorem An equation of the circle with center (a,b) and radius r is r 2 = (x a) 2 + (y-b) 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 (x-1) 2 + (y-2) 2 = 9. (1, -2), r = 3 SECTION 13-2 Slope of a Line FORMULA FOR SLOPE m = change in y-coordinate change in x-coordinate 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 13-3 SECTION 13-3 Parallel and Perpendicular Lines Two nonvertical lines are parallel if and only if their slopes are equal 3

4 Theorem 13-4 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 13-4 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 13-5 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 13-6 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 13-6 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 Y-intecept = C/B 8

9 Theorem 13-7 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 13-8 Point-Slope Form Writing Linear Equations SECTION 13-7 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