13.1 The Distance Formulas

Review of Graphs

Coordinate Plane

Origin

Axes: x-axis y-axis

The arrowhead on each side represents the positive direction

Quadrants   VV C

Remote Time Which quadrant does the point lie in?

Remote Time (3,1)

Remote Time (-2,-1)

Remote Time (5,-3)

Finding Distance A(3,4)B(-1,4) D(-3,-2) C(-3,1) **Distance is always a positive Number**

Horizontal Distance AB = |3-(-1)| = 4 or AB = |-1-3| = 4 A(3,4)B(-1,4) **A horizontal line will always have y-coordinates that are the same** What do you notice about the coordinates of each points?

Vertical Distance CD = |1-(-2)|= 3 or AB = |-2-1| = 3 D(-3,-2) C(-3,1) **A vertical line will always have x-coordinates that are the same**

What do I do when two points do not lie on a horizontal or vertical line……..? A(1,2) B(4,-2) PARTNERS: Come up with a logical way in which you think you could find the distance of your slanted line.

Make a right triangle A(1,2) B(4,-2) C(1,-2)

Use the pythagorean theorem A(1,2) B(4,-2) C(1,-2)

Theorem The Distance Formula: The distance d between two points is given by

Find the distance 1.(0,0) (0,-6) 2.(-3,-8) (2,4) 3.(9,-4) (1,8) √5

The equation of a circle What would happened if I squared both sides of the distance formula? d 2 = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 The equation of a circle with center (h, k) and a radius (r) is… r 2 = (x – h) 2 + (y – k) 2

3 different ways to use equation of a circle 1.Write an equation given the center and radius 2.What are the coordinates of the center and the radius, given the equation 3.Write the equation given the center and a point

Whiteboards Brightstorm examples Classroom exercises pg. 526 –#11 b –#11 a –#12 b

13.2 Slope of a Line

Coach Hud Chad Eggertson Forrest

Slope (m) Steepness of a line

Slope: Pick 2 points on the line (x 1, y 1 ) (x 2, y 2 ) Does it matter which y you start with?

Partner Practice Find the slope of the line through (-3,4) and (-4,5) Slope = - 1

4 Types of Slope

“Read” the line Positive Negative Partners: Come up with 2 points that would give me a positive slope and 2 points that would give me a negative slope.

No Slope Zero Slope Find the slope of CD: C (2,1) D (2, 7) Find the slope of AB: A (1,2) B (5,2)

No Slope (undefined) Zero Slope

No Slope (undefined) EC: Which axis has an undefined slope?

Zero Slope

White Board Practice Find the slope of the line through (6,-3) and (-1,-2). Then Graph it. - 1/7 Find the slope of the line through (8,-4) and (-3,-4). Then graph it. 0

White Board Practice Sketch the line and find the coordinates of two other points on the line. –The line goes through P(-2,1) and has a slope = 1 / 3

White Board Practice Sketch the line and find the coordinates of two other points on the line. –The line goes through P(2,4) and has a slope =

Remote Time A: Always B: Sometimes C: Never D: I don’t know

A: Always B: Sometimes C: Never D: I don’t know The slope of a vertical lines is ___________ zero.

A: Always B: Sometimes C: Never D: I don’t know The slope of a horizontal line is ___________ zero.

A: Always B: Sometimes C: Never D: I don’t know The slope of a line that rises to the right is __________ positive

A: Always B: Sometimes C: Never D: I don’t know The slope of a line that falls to the right is __________ negative

13.3 Parallel and Perpendicular Lines

Graph the following –Line AB A(1, -5) B (4,0) –A line with point at D (0, -3) and m = 5/3

Parallel Lines Coplanar lines that do not intersect

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

Partners Give me the coordinates of 2 points for one line and 2 points for another line I want the lines to be parallel with negative slopes

Perpendicular lines

Graph the following –Line AB A(-1, 1) B (4,-2) –A line with point at D (0, -3) and m = 5/3

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

In English… That means that the slopes of two perpendicular lines are –Opposite (one + and one -) –Reciprocals ( and )

White Board Practice r || s and r  t Slope of rSlope of sSlope of t

White Board Practice r || s and r  t Slope of rSlope of sSlope of t

White Board Practice r || s and r  t Slope of rSlope of sSlope of t

White Board Practice r || s and r  t Slope of rSlope of sSlope of t

White Board Practice r || s and r  t Slope of rSlope of sSlope of t

White Board Practice r || s and r  t Slope of rSlope of sSlope of t

Remote Time The slopes of two lines are given. Are the lines A: Parallel B: Perpendicular C: Neither D: I don’t know

A: Parallel B: Perpendicular C: Neither D: I don’t know

White Board Practice Use slopes to show that a quadrilateral with vertices A(-2,7), B(3,7), C(6,11), and D(1,11) is a parallelogram.

Consider triangle ABC where A=(1,1) B=( 5,2 ) C=( 4,6 ) Do a quick sketch. How could we prove that this triangle is an isosceles right triangle? We’d need to: –Show two sides were congruent using the Distance Formula –Show that one of the angles is 90 degrees by Showing two segments are perpendicular (have slopes that are opposite reciprocals)

Consider triangle ABC where A=(1,1) B=( 5,2 ) C=( 4,6 ) Now Do It! Using the slope and distance formulas show that ABC is a right isosceles triangle.

13-5 Midpoint Formula

Warm – up In your own words, write down what the midpoint formula is… –What is it allowing us to find? –How is the answer going to be given?

4 facts from ch 13 Parallel lines have the same slope. Perpendicular lines have opposite reciprocal slopes (2/3 vs. -3/2)

A B C Name the midpoint of segment AB. Name the midpoint of segment AC. Describe your strategy. What does the midpoint of BC appear to be? Can you adapt your strategy above to find it algebraically?

Summary: To find the midpoint of a segment, simply average the x coordinates and average the y coordinates. OR: The midpoint between Is:_______________

3 types of midpoint problems 1.Find the midpoint of segment Find midpoint of AB if A=(-3, 7) and B=(9, -20) 1.Midpoint = (3, -6.5) 2.Find the other endpoint when given the midpoint (3, -1) is the midpoint of segment RS. If R = (12, -5) find the coordinates of S. –S (-6, 3)

3. Show a point is on the perpendicular bisector of a segment Given points A(5,2) and B(3,-6) show that P(0,-1) is on the perpendicular bisector. Draw a sketch to help. –Find the midpoint of AB –Find slope of AB and slope of PM –If the product of slope AB and slope PM = -1, then P is on the perpendicular bisector of AB.

Whiteboards Finding other endpoint ebra-2/graphs-of-linear- equations/midpoint-formula-problem- 2/ ebra-2/graphs-of-linear- equations/midpoint-formula-problem- 2/

A BC Show that point C is on the perpendicular bisector of line AB.

Challenge: F=(-2,-4) G=(4,-1) H=(1,1) I=(-1,0) Sketch and label A) Use the slope formula to show that FGHI is a trapezoid B) Use the midpoint and slope formulas to show that the median of the trapezoid (the segment that connects the midpoints of the legs) is parallel to both bases.

13.6 Graphing Linear Equations

What is a Linear Equation? A linear equation is an equation whose graph is a LINE. Linear Not Linear

Does the point (4,2) lie on the line x + y = 6 ?

Forms of Linear Equations The forms of linear equations are the formats in which the information is written in. These two forms are the most commonly used ways to write linear equations. 1. Standard Form: Ax + By =C 2. Slope Intercept Form: y=mx+b

Standard Form Standard form for linear equations is Ax + By=C. It can be used to find two points on the line of the equation. **More specifically the x and y int.** Example: 2x+2y=4 1.) Substitute in a zero for x. Simplify. 2(0)+2y=4 y=2 2.) One point of the line is (0,2). Plot the point. 3.) Substitute y with zero. Simplify. 2x+2(0)=4 x=2 4.) The second point is (2,0). Plot it and draw a line through the two points. x y

Slope Intercept Form *Get y by itself* Slope intercept form is y=mx+b. This form makes it easy to find the slope (m) and the y-intercept (b). Working with this form is simple, so it is used more often than other forms. Example: y= ¾x + 3 * ¾ is the slope. * 3 is the point where the line crosses the Y-axis.

Graph Using Slope Intercept Form I will use the equation y= ½ x-1 to demonstrate how to graph a linear equation using slope intercept form. 1.) First, graph the y-intercept (b). In this equation, b= -1, so place a dot on the point (0,-1). 2.) Next, use the slope (½ ) to rise up once and run twice horizontally. Plot the point. 3.) Plot as many points as desired using the slope, then draw a line through the points. Rise 1 Run 2

Graphing a Linear Equation Let’s try x – 2y = 6. First Step: Write y as a function of x x – 2y = 6 –2y = 6 – x y = 1/2x - 3 3

Graphing Horizontal & Vertical Lines When you are asked to graph a line, and there is only ONE variable in the equation, the line will either be vertical or horizontal. For example … Graph x = 3 Since there are no y – values in this equation, x is always 3 and y can be any other real number. x = 3 Graph y = –2 Since there are no x – values in this equation, y is always – 2 and x can be any other real number. y = –2 THINK OPPOSITE OF THE AXIS!!

Solving System of Equations 2x + 3y = 18 2x - y = 10 2 different ways to solve… 1.Substitution Method 2.Add/subtract method PARTNERS: find the value of x and y using one of the methods. **What is the x and y values that will work for both equations? (In other words….where on the coordinate plane do the two lines intersect?)

Whiteboards Page 69: –#4, –11, –18 Pg. 550 (classroom exercises) –#5 –#11

13.7 Writing Linear Equations

WARM -UP Use pg. 553 to help answer to following… Put the equation of the line in slope intercept form 1.m = -5/3 ; y-int = 4 2.x-int = -6 ; y-int = 3

Forms of Linear Equations 1.Standard Form: Ax + By =C 2.Slope Intercept Form: y=mx+b 3.Point-Slope Form: y – y 1 = m(x-x 1 ) **Always simplify the final answer by putting it in slope-intercept form.**

Point Slope Form Best used for writing the equation of a line when given information about the line. It will helps us get to the point where we can write the equation in slope int. form An equation of the line that passes through the point (x 1, y 1 ) and has slope m. y – y 1 = m(x - x 1 )

Find an equation of the line described Ex1. The line with slope 2 containing the point (5,-2) y – y1 = m(x - x1) y + 2 = 2(x – 5)  insert m and the point y + 2= 2x – 10  distribute the 2 to the x and -5 y= 2x – 12  get y by itself

Find an equation of the line described Ex 2. The line through (3,-2) and (4,1) -2 – 1 m = 3  Find the slope 3-4 y + 2 = 3(x-3)  plug in m and 1 point y = 3x – 11  slope int form

Find an equation of the line described Ex3. The line with x-intercept 5 and y-intercept –3. y = 3/5x - 3

Find an equation of the line described Ex4. The line through (6,4) and perpendicular to the line y = -2x+4 y – 4 = ½(x - 6) y = 1/2x + 1

Find an equation of the line described Ex 5. A vertical line going through (5,-3) x = 5

Find an equation of the line described Ex 5. A horizontal line going through (-4,7) y = 7

Chapter 13.8 Organizing Coordinate Proofs

Review Squares Four congruent sides Four Right angles

Review Rectangles Opposite sides are congruent Four Right angles

Review Parallelograms Opposite sides are congruent Opposite angles are congruent Opposite sides are parallel

Review Isosceles Triangles Legs are congruent Base angles are congruent

Review Isosceles Trapezoids Legs are congruent Base angles are congruent

Review Rhombus All sides are congruent

Quick Review In an ordered pair which one is the x-coor and which is the y-coor? The x-coordinate tells us to do what ? The y-coordinate tells us to do what? (3,-7)

Rectangle ( 4,3 ) (, )

Rectangle ( a,b ) (, )

Equilateral Triangle (, ) ( 3, 5 ) (, )

Equilateral Triangle (, ) ( a, b ) (, )

Rhombus ( c, 0 ) ( a, b ) (, )

Square ( 3k, 0 ) (, )

Isosceles Triangle ( 2a, 0 ) (, ) (, b )

Isosceles Trapezoid ( 3e, 0 ) ( -3e,0 ) ( 2e, f ) (, )

Isosceles Right Triangle ( s, 0 )(, )

Parallelogram ( i, 0 ) ( h, ) (, g )