CHAPTER 13 Geometry and Algebra

SECTION 13-1 The Distance Formula

The distance between two points (x1, y1) and (x2, y2) is given by: Theorem 13-1 The distance between two points (x1, y1) and (x2, y2) is given by: D = [(x2 – x1)2 + (y2-y1)2]½

Example Find the distance between points A(4, -2) and B(7, 2) d = 5

An equation of the circle with center (a,b) and radius r is 13-2 Theorem An equation of the circle with center (a,b) and radius r is r2 = (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

SLOPE is the ratio of vertical change to the horizontal change. The variable m is used to represent slope.

FORMULA FOR SLOPE Or m = rise run m = change in y-coordinate change in x-coordinate Or m = rise run

SLOPE OF A LINE m = y2 – y1 x2 – x1

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

Lines with positive slope rise to the right. 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

Parallel and Perpendicular Lines SECTION 13-3 Parallel and Perpendicular Lines

Theorem 13-3 Two nonvertical lines are parallel if and only if their slopes are equal

Theorem 13-4 Two nonvertical lines are perpendicular if and only if the product of their slopes is - 1

Find the slope of a line parallel to the line containing points M and N.

Find the slope of a line perpendicular to the line containing points M and N.

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

SECTION 13-4 Vectors

DEFINITIONS Vector– any quantity such as force, velocity, or acceleration, that has both size (magnitude) and direction

Vector AB 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 

Use the Pythagorean Theorem or the Distance Formula to find the magnitude of a vector.

Given: Points P(-5,4) and Q(1,2) EXAMPLE Given: Points P(-5,4) and Q(1,2) Find PQ Find  PQ 

In general, if the vector PQ = (a,b) Scalar Multiple In general, if the vector PQ = (a,b) then kPQ = (ka, kb)

Vectors having the same magnitude and the same direction. 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.

Parallel Vectors Two vectors are parallel if the arrows representing them have the same direction or opposite directions.

Determine whether (6,-3) and (-4,2) are parallel or perpendicular. EXAMPLE Determine whether (6,-3) and (-4,2) are parallel or perpendicular.

Determine whether (6,-3) and (2,4) are parallel or perpendicular. EXAMPLE Determine whether (6,-3) and (2,4) are parallel or perpendicular.

Adding Vectors (a,b) + (c,d) = (a+c, b+d)

Find the Sum Vector PQ = (4, 1) and Vector QR = (2, 3). Find the resulting Vector PR.

SECTION 13-5 The Midpoint Formula

Midpoint Formula M( x1 + x2, y1 + y2) 2 2

Example Find the midpoint of the segment joining the points (4, -6) and (-3, 2) M(1/2, -2)

Graphing Linear Equations SECTION 13-6 Graphing Linear Equations

is an equation whose graph is a straight line. LINEAR EQUATION is an equation whose graph is a straight line.

The graph of any equation that can be written in the form 13-6 Standard Form The graph of any equation that can be written in the form Ax + By = C Where A and B are not both zero, is a line

Example Graph the line 2x – 3y = 12 Find the x-intercept and the y-intercept and connect to form a line

The slope of the line Ax + By = C (B ≠ 0) is - A/B Y-intercept = C/B THEOREM The slope of the line Ax + By = C (B ≠ 0) is - A/B Y-intercept = C/B

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

Writing Linear Equations SECTION 13-7 Writing Linear Equations

Theorem 13-8 Point-Slope Form An equation of the line that passes through the point (x1, y1) and has slope m is y – y1 = m (x – x1)

Write an equation of a line with the given slope and through a given point m=-2 P(-1, 3)

Write an equation of a line with the through the given points (2, 5) (-1, 2)

Write an equation of a line through (6, 4) and parallel to the line y = -2x +4

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