Write an equation given the following info: Warm up Write an equation given the following info: 1. m = (-9, -1) 2. (-2, -1) (-2, 3) 3. (-2, 6) (2, 8) 4. m = 0 (3, 4)

5. & 6. Write the equation of each graph     y = 2x – 3 y = -3/2x + 3

Unit 2: Geometric & Algebraic Connections LG 2-1 Coordinate Proofs LG 2-2 Equations of Circles LG 2-3 Geometric Modeling TEST February 7th!

LG 2-1 Coordinate Proofs: write the equations of parallel and perpendicular lines find the point that partitions a line segment in a given ratio use coordinate algebra to determine perimeter and area of defined figures ESSENTIAL QUESTIONS How can a line be partitioned? How are the slopes of lines used to determine if the lines are parallel, perpendicular, or neither? How can slope and the distance formula be used to determine properties of polygons and classify them?

SELECTED TERMS AND SYMBOLS Distance Formula Formula for finding the point that partitions a directed segment AB at the ratio of a : b from A(x1, y1) to B(x2, y2):

Writing Equations of Lines

I. How to Write an Equation of a Line Given m and b 1. Write down y = mx + b 2. Substitute slope for m and y-intercept for b. 3. Simplify the equation

Write the equation of the line given m and b. Ex. 1 Slope is -5 and y-intercept is 2 y = -5x + 2 Ex. 2 Slope is -1/2 and y-intercept is -2 y = -½x – 2

Write the equation of the line given m and b. Ex. 3 Slope is 0 and y-intercept is 3 y = 3 Ex. 4 Slope is 1/3 and y-intercept is 0

II. How to Write an Equation of a Line Given a Graph Write down y = mx + b Use any 2 “good” points on the graph to find the slope, m. Find the y-intercept on the graph, b. Substitute slope for m and y-int for b into the equation y = mx + b.

5. Write the equation of this graph  

6. Write the equation of this graph  

7. Write the equation of this graph   y = -3x + 2

8. Write the equation of this graph   y = 2x

9. Write the equation of this graph   y = 3

10. Write the equation of this graph x = - 6

11. & 12. Write the equation of each graph     y = 2x – 3 y = -3/2x + 3

III. How to Write an Equation of a Line Given m and a point Write down y = mx + b. Substitute slope for m and the point (x, y). Solve for b. Substitute m and b back into the equation.

Write the equation of the line given m and a point Ex 13: m = 2 Point: (2, 3) y = mx + b 3 = 2 (2) + b b = -1 y = 2x – 1

Write the equation of the line given m and a point Ex 14: m = 1/2 Point: (4,-3) y = mx + b -3 = 1/2 (4) + b b = -5 y = ½ x – 5

Ex 15: m = -2 Point: (-5, 3) y = mx + b 3 = -2 (-5) + b b = -7 Write the equation of the line given m and a point Ex 15: m = -2 Point: (-5, 3) y = mx + b 3 = -2 (-5) + b b = -7 y = -2x – 7

Write the equation of the line given m and a point Ex: 16 m = 4 (1,4) y = 4x

Write the equation of the line given m and a point Ex: 17 m = ½ (-1,-2) y = ½ x – 1 ½

Write the equation of the line given m and a point Ex: 18 m = 2 (0,3) y = 2x + 3

Write the equation of the line given m and a point Ex: 19 m = 3 (3,0) y = 3x – 9

Write the equation of the line given m and a point Ex: 20 m = undefined (3,6) x = 3

IV. How to Write an Equation of a Line Given TWO points Write down y = mx + b. Use the slope formula to find m. Pick one of the ordered pairs & substitute slope for m and the point (x, y). Solve for b. Substitute m and b into the equation.

Equation of a Line - Given 2 points Ex: 21 (2, 3) (4, 5) y = mx + b 3 = 1(2) + b b = 1 y = x + 1

Equation of a Line - Given 2 points Ex: 22 (2, 3) (-4, 15) y = -2x + 7

Equation of a Line - Given 2 points Ex: 23 (2, 2) (0, 4) y = -x + 4

Equation of a Line - Given 2 points Ex: 24 (2,3) (1,4) y = -x + 5

Equation of a Line - Given 2 points Ex: 25 (4,5) (5,2) y = -3x + 17

Graphs: Lines Never Intersect and are in the same plane PARALLEL LINES Graphs: Lines Never Intersect and are in the same plane Equations: Same Slopes Different y-intercepts

How to Write an Equation of a Line PARALLEL to another and given a point 1. Given equation should be solved for y (y = mx + b) Write down the slope of that line Substitute m and (x, y) in y = mx + b. Solve for b. Write the equation using m and b.

Write a line parallel to the line 2x + y = 3 and passes through the point (-2, 5). Work on left side of t-chart, step by step notes on right side y = -2x + 1

Write a line parallel to the line y = 3x – 5 and passes through the point (-5, -2). Work on left side of t-chart, step by step notes on right side y = 3x + 13

Write a line parallel to the line y = -4x + 1 and passes through the point (2, -1). Work on left side of t-chart, step by step notes on right side y = -4x + 7

Write a line parallel to the line y = -x – 7 and passes through the point (-4, -4). Work on left side of t-chart, step by step notes on right side y = -x – 8

Graphs: Lines Intersect at 90º angles PERPENDICULAR LINES Graphs: Lines Intersect at 90º angles Equations: Opposite Reciprocal Slopes With the same or different y-int

Find the Opposite Reciprocal Slopes

Are these lines parallel, perpendicular, or neither? y = -2x + 1 y = -2x – 4 y = 3x – 4 y = -3x + 1 y = 1/5 x + 2 y = -5x + 6 parallel neither perpendicular

Are these lines parallel, perpendicular, or neither? 4. y = -2x + 1 y = -1/2x – 4 5. y = 3x – 4 y = 1 + 3x 6. y = 5/6 x + 2 y = -6/5 x + 6 neither parallel perpendicular

1. Given equation should be solved for y (y = mx + b) How to Write an Equation of a Line PERPENDICULAR to another and given a point 1. Given equation should be solved for y (y = mx + b) Write down the OPPOSITE RECIPROCAL slope of that line Substitute m and (x, y) in y = mx +b. Solve for b. Write the equation using m and b.

Write a line perpendicular to the line y = ½ x – 2 and passes through the point (1, 0). Work on left side of t-chart, step by step notes on right side y = -2x + 2

Write a line perpendicular to the line y = -3x + 2 and passes through the point (6, 5). Work on left side of t-chart, step by step notes on right side y = 1/3x + 3

Write a line perpendicular to the line 2x + 3y = 9 and passes through the point (6, -1). Work on left side of t-chart, step by step notes on right side y = 3/2 x – 10

Write a line perpendicular to the line y = 2x – 1 and passes through the point (2, 4). Work on left side of t-chart, step by step notes on right side y = -1/2x + 5

Write a line perpendicular to the line and passes through the point (5, 1). Work on left side of t-chart, step by step notes on right side y = 3x – 14

Determine whether the given lengths are sides of a right triangle. Warm-up Use the Pythagorean theorem to find the missing length of the right triangle. Round to the nearest tenth. 1. 2. Determine whether the given lengths are sides of a right triangle. 3. 8, 15, 17 4. 3, 6, 7 5. 9, 40, 41 c 2 17 8 5 b

Write the following equations: A line parallel and a line perpendicular to the line 3x – 2y = 8 through the point (-2,1). A line parallel and a line a line perpendicular to the line x = 7 through the point (3, 8). A line parallel and a line perpendicular to the line y = 7 through the point (-4, 0).

The Distance Formula (x2,,y2) d (x1,,y1) (x2,,y1)

Warm UP

The Distance Formula

Steps to solving the Distance Formula Write the distance formula Substitute Simplify Evaluate Powers Add Use a calculator

The Distance Formula (3,4) (1,1)

Example #1 Use the distance formula to find the distance between the points, (10, 5) and (40, 45). D = 50

Use the distance formula to find the distance Example #2 Use the distance formula to find the distance between (1, 4) and (-2, 3) D = 3.16

The midpoint between (x1, y1) and (x2, y2) is The Midpoint Formula The midpoint of a line segment is the point on the segment that is equidistant from its endpoints The midpoint between (x1, y1) and (x2, y2) is

Example 1 Find the midpoint of the line segment connecting the given points. (-2, 3) and (4, 2)

Example 2 Find the midpoint of the line segment connecting the given points. (200, 75) and (25, 175).

Find the midpoint of the line segments. Example Find the midpoint of the line segments.

Partition Line Segments (1 Dimension) A is at x1, and B is at x2. Find the point, T, so that T partitions A to B in an a:b ratio.

1. A is at 1, and B is at 7. Find the point, T, so that T partitions A to B in a 2:1 ratio.

2. A is at -6 and B is at 4. Find the point, T, so that T partitions A to B in a 2:3 ratio.

Partition Line Segments (2 Dimensions) A is at (x1, y1) and B is at (x2, y2) Find the point, T, so that T partitions A to B in an a:b ratio.

Ex 1: Find the coordinates of P along the directed line segment AB so that the ratio of AP to PB is 3 to 2. (4.8, 7.6)

Ex 2: Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(1, 3), B(8, 4); 4 to 1. (6.6, 3.8)

EX 3: Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(-2, 1), B(4, 5); 3 to 7. (-0.2, 2.2)

Ex 4: Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(8, 0), B(3, -2); 1 to 4. (7, -0.4)

Review Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. A(-2, -4), B(6, 1); 3 to 2. Find the coordinates of point P along the directed line segment BA in a 1:4 ratio. A(8, 0), B(3, -2)

Connecting Geometric & Algebraic Concepts: The goal of this assignment is to use the distance and slope formulas to prove statements about geometric figures on the coordinate plane. Since the purpose is to prove a statement, you must show work.

Quadrilateral 1: Plot and label each point Quadrilateral 1: Plot and label each point. A(-5, 6), B(3, 7), C(4, -1), and D(-4, -2). Definition: A parallelogram is a quadrilateral with two pairs of opposite sides that are parallel. Using the definition of parallelogram, prove that Quadrilateral 1 is a parallelogram

Quadrilateral 1: Plot and label each point Quadrilateral 1: Plot and label each point. A(-5, 6), B(3, 7), C(4, -1), and D(-4, -2). Theorem: A parallelogram with four right angles is a rectangle. Using the theorem, prove that Quadrilateral 1 is a rectangle.

Quadrilateral 1: Plot and label each point Quadrilateral 1: Plot and label each point. A(-5, 6), B(3, 7), C(4, -1), and D(-4, -2). Definition: A rhombus is a parallelogram with all sides congruent. Using the definition, prove that Quadrilateral 1 is a rhombus.

Quadrilateral 1: Plot and label each point Quadrilateral 1: Plot and label each point. A(-5, 6), B(3, 7), C(4, -1), and D(-4, -2). Definition: A square is a rectangle and a rhombus. Using the definition, is Quadrilateral 1 a square? Why?

Quadrilateral 1: Plot and label each point Quadrilateral 1: Plot and label each point. A(-5, 6), B(3, 7), C(4, -1), and D(-4, -2). Theorem: The diagonals in a rhombus are perpendicular. Prove that the theorem is true for Quadrilateral 1.

Continue working on the packet 

Warm UP

LG 2-2 Equations of Circles: Develop equations from geometric definition of circles address equations of circles in standard and general forms graph circles by hand and by using graphing technology ESSENTIAL QUESTIONS How can I use the Pythagorean Theorem to derive the equation of a circle? How are the graph of a circle and its equation related? How are the equation of a circle and its graph related? How can I prove properties of geometric figures algebraically?

SELECTED TERMS AND SYMBOLS Center of a Circle: The point inside the circle that is the same distance from all of the points on the circle. Standard Form of a Circle: where (h,k) is the center and r is the radius.

For this task, you need a straightedge (a ruler), a PENCIL and a calculator.

Graphing and Writing Equations of Circles

Standard Form of a Circle r is the radius of the circle Center is at (h, k) r is the radius of the circle

EX 1 Write an equation of a circle with center (3, -2) and a radius of 4. k r

EX 2 Write an equation of a circle with center (-4, 0) and a diameter of 10. k 2r

EX 3 Write an equation of a circle with center (2, -9) and a radius of . k r

EX 4 Find the coordinates of the center and the measure of the radius. Opposite signs! ( , ) 6 -3 Take the square root! Radius 5

5. Find the center, radius, & equation of the circle. The center is The radius is The equation is (0, 0) 12 x2 + y2 = 144

6. Find the center, radius, & equation of the circle. (1, -3) The center is The radius is The equation is 7 (x – 1)2 + (y + 3)2 = 49

7. Graph the circle, identify the center & radius. (x – 3)2 + (y – 2)2 = 9 Center (3, 2) Radius of 3     

Warm UP 1) Write the equation of a circle that passes through (2, 3) and has center Q(2, –1) 2) Write the equation of a circle that passes through (–2, –1) and has center D(2, –4).

General Form of a Circle

General Form of a Circle Every binomial squared has been multiplied out. Every term is on the left side, equal to 0. Squared terms go first in alpha order.

Converting from General to Standard A needs to be 1. Divide if needed. Move the x terms together and the y terms together. Move E to the other side of the equals sign. Complete the square (as needed) for x. Complete the square(as needed) for y. Factor the left & simplify the right.

8. Write the standard equation of the circle. State the center & radius.

9. Write the standard equation of the circle. State the center & radius.

10. Write the standard equation of the circle 10. Write the standard equation of the circle. State the center & radius.

11. Write the general form of the equation of the circle.

Review Test

NOPE! Warm UP! 1) Write the equation of the circle: (x – 2)2 + (y + 4)2 = 16 NOPE! 3) A point on the circle has coordinates (3,y). Find a value for y. y = 4 or -4

In your own words, compare and contrast the following circles In your own words, compare and contrast the following circles. Write in complete sentences. Turn this in when you are finished.

Write equations for the following circles (if you didn’t already) Write equations for the following circles (if you didn’t already). What do you call the FORM of the equations you wrote?

ODD ONE OUT! Each of these three circles might be considered to be the odd one out. With a partner, choose one and write why it might be the odd one out. What properties have the other two circles got in common that the third does not have?

Write the NUMBER of the card on the placemat  group Write the NUMBER of the card on the placemat  DO NOT WRITE ON THE CARDS PLEASE!

On the back of your quiz: Extra Credit On the back of your quiz:

Warm UP!

QUIZ TIME

Prove or disprove that the point A(10, 3) lies on a circle centered at C(5, -2) and passing through the point B(6, 5).

To add points to your quiz yesterday, complete the following To add points to your quiz yesterday, complete the following. Each question replaces one you missed. Write the equation of the line through the point (3,3) for the line perpendicular to y = -1/3x + 5.   2. Write the equation of the line through the point (0, 7) for the line parallel to 3x + 4y = 12. 3. Given the directed segment from A(5,4) to B(0, 14) partition the segment with point P in a ratio of 2:3.

Warm UP:

LG 2-3 Geometric Modeling: use Algebra to model Geometric ideas Solve real world problems that can be modeled using density, area, and volume concepts. Model everyday objects using three dimensional shapes and describe the object using characteristics of the shape. ESSENTIAL QUESTIONS How can I minimize cost and maximize the volume of a topless box?

LG 2-3 Geometric Modeling Geometric Modeling is the field that discusses the mathematical methods behind the modeling of realistic objects. One of the main applications of this field is for computer graphics and computer aided design. The field was initially developed in the mid-1970s and has evolved as the speed and memory of computer systems has advanced.

Find the intersection of the two equations:

Josh and Drake decide to play catch after school Josh and Drake decide to play catch after school. They start at the same point. Josh walks 50 feet north and 20 feet west. Drake walks 40 feet south and 10 feet east. How far apart are they? Joe’s Pizza Parlor advertises free delivery within a 15 mile radius. If a customer lives 12 miles east and 10 miles south of Joe’s, do they qualify for free delivery? A given circle has a radius of 6 and is centered at the origin. The line x = 3 intercepts the circle and forms a chord. What is the length of the chord? Round your answer to the nearest tenth.