Please read the following and consider yourself in it. An Affirmation Please read the following and consider yourself in it. I am capable of learning. I can accomplish mathematical tasks. I am ultimately responsible for my learning.

Standards MGSE9-12.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. (Focus on quadrilaterals, right triangles, and circles.) MGSE9-12.G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). MGSE9-12.G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. MGSE9-12.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Objectives MGSE9-12.G.GPE.5 SWBAT Prove the slope criteria for parallel and perpendicular lines IOT solve geometric problems. SWBAT Apply slope criteria IOT synthesize the equation of a line parallel or perpendicular to a given line that passes through a given point.

What is the slope of the line MN for M(–3, 4) and N(5, –8)? B. C. D. 5-Minute Check 1

What is the slope of the line MN for M(–3, 4) and N(5, –8)? B. C. D. 5-Minute Check 1

What is the slope of a line perpendicular to MN for M(–3, 4) and N(5, –8)? B. C. D. 5-Minute Check 2

What is the slope of a line perpendicular to MN for M(–3, 4) and N(5, –8)? B. C. D. 5-Minute Check 2

What is the graph of the line that has slope 4 and contains the point (1, 2)? A. B. C. D. 5-Minute Check 4

What is the graph of the line that has slope 4 and contains the point (1, 2)? A. B. C. D. 5-Minute Check 4

What is the graph of the line that has slope 0 and contains the point (–3, –4)? A. B. C. D. 5-Minute Check 5

What is the graph of the line that has slope 0 and contains the point (–3, –4)? A. B. C. D. 5-Minute Check 5

Mathematical Practices 4 Model with mathematics. Content Standards G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). Mathematical Practices 4 Model with mathematics. 8 Look for and express regularity in repeated reasoning. CCSS

slope-intercept form point-slope form Vocabulary

Concept

y = mx + b Slope-intercept form y = 6x + (–3) m = 6, b = –3 Slope and y-intercept Write an equation in slope-intercept form of the line with slope of 6 and y-intercept of –3. Then graph the line. y = mx + b Slope-intercept form y = 6x + (–3) m = 6, b = –3 y = 6x – 3 Simplify. Example 1

Slope and a Point on the Line Write an equation in point-slope form of the line whose slope is that contains (–10, 8). Then graph the line. Point-slope form Simplify. Example 2

Graph the given point (–10, 8). Slope and a Point on the Line Answer: Graph the given point (–10, 8). Use the slope to find another point 3 units down and 5 units to the right. Draw a line through these two points. Example 2

Graph the given point (–10, 8). Slope and a Point on the Line Answer: Graph the given point (–10, 8). Use the slope to find another point 3 units down and 5 units to the right. Draw a line through these two points. Example 2

Write an equation in point-slope form of the line whose slope is that contains (6, –3). B. C. D. Example 2

Write an equation in point-slope form of the line whose slope is that contains (6, –3). B. C. D. Example 2

Step 1 First, find the slope of the line. Two Points A. Write an equation in slope-intercept form for a line containing (4, 9) and (–2, 0). Step 1 First, find the slope of the line. Slope formula x1 = 4, x2 = –2, y1 = 9, y2 = 0 Simplify. Example 3

Distributive Property Two Points Step 2 Now use the point-slope form and either point to write an equation. Point-slope form Using (4, 9): Distributive Property Add 9 to each side. Answer: Example 3

Distributive Property Two Points Step 2 Now use the point-slope form and either point to write an equation. Point-slope form Using (4, 9): Distributive Property Add 9 to each side. Answer: Example 3

Step 1 First, find the slope of the line. Two Points B. Write an equation in slope-intercept form for a line containing (–3, –7) and (–1, 3). Step 1 First, find the slope of the line. Slope formula x1 = –3, x2 = –1, y1 = –7, y2 = 3 Simplify. Example 3

Distributive Property Two Points Step 2 Now use the point-slope form and either point to write an equation. Point-slope form Using (–1, 3): m = 5, (x1, y1) = (–1, 3) Distributive Property Add 3 to each side. y = 5x + 8 Answer: Example 3

Distributive Property Two Points Step 2 Now use the point-slope form and either point to write an equation. Point-slope form Using (–1, 3): m = 5, (x1, y1) = (–1, 3) Distributive Property Add 3 to each side. y = 5x + 8 Answer: Example 3

This is a horizontal line. Write an equation of the line through (5, –2) and (0, –2) in slope-intercept form. Step 1 Slope formula This is a horizontal line. Example 4

Subtract 2 from each side. y = –2 Horizontal Line Step 2 Point-Slope form m = 0, (x1, y1) = (5, –2) Simplify. Subtract 2 from each side. y = –2 Answer: Example 4

Subtract 2 from each side. y = –2 Horizontal Line Step 2 Point-Slope form m = 0, (x1, y1) = (5, –2) Simplify. Subtract 2 from each side. y = –2 Answer: Example 4

Concept

y = mx + b Slope-Intercept form 0 = –5(2) + b m = –5, (x, y) = (2, 0) Write Equations of Parallel or Perpendicular Lines y = mx + b Slope-Intercept form 0 = –5(2) + b m = –5, (x, y) = (2, 0) 0 = –10 + b Simplify. 10 = b Add 10 to each side. Answer: Example 5

y = mx + b Slope-Intercept form 0 = –5(2) + b m = –5, (x, y) = (2, 0) Write Equations of Parallel or Perpendicular Lines y = mx + b Slope-Intercept form 0 = –5(2) + b m = –5, (x, y) = (2, 0) 0 = –10 + b Simplify. 10 = b Add 10 to each side. Answer: So, the equation is y = –5x + 10. Example 5

A. y = 3x B. y = 3x + 8 C. y = –3x + 8 D. Example 5

A. y = 3x B. y = 3x + 8 C. y = –3x + 8 D. Example 5

A = mr + b Slope-intercept form A = 525r + 750 m = 525, b = 750 Write Linear Equations RENTAL COSTS An apartment complex charges $525 per month plus a $750 annual maintenance fee. A. Write an equation to represent the total first year’s cost A for r months of rent. For each month of rent, the cost increases by $525. So the rate of change, or slope, is 525. The y-intercept is located where 0 months are rented, or $750. A = mr + b Slope-intercept form A = 525r + 750 m = 525, b = 750 Answer: Example 6

A = mr + b Slope-intercept form A = 525r + 750 m = 525, b = 750 Write Linear Equations RENTAL COSTS An apartment complex charges $525 per month plus a $750 annual maintenance fee. A. Write an equation to represent the total first year’s cost A for r months of rent. For each month of rent, the cost increases by $525. So the rate of change, or slope, is 525. The y-intercept is located where 0 months are rented, or $750. A = mr + b Slope-intercept form A = 525r + 750 m = 525, b = 750 Answer: The total annual cost can be represented by the equation A = 525r + 750. Example 6

Evaluate each equation for r = 12. Write Linear Equations RENTAL COSTS An apartment complex charges $525 per month plus a $750 annual maintenance fee. B. Compare this rental cost to a complex which charges a $200 annual maintenance fee but $600 per month for rent. If a person expects to stay in an apartment for one year, which complex offers the better rate? Evaluate each equation for r = 12. First complex: Second complex: A = 525r + 750 A = 600r + 200 = 525(12) + 750 r = 12 = 600(12) + 200 = 7050 Simplify. = 7400 Example 6

Write Linear Equations Answer: The first complex offers the better rate: one year costs $7050 instead of $7400. Example 6

A. Write an equation to represent the total cost C for d days of use. RENTAL COSTS A car rental company charges $25 per day plus a $100 deposit. A. Write an equation to represent the total cost C for d days of use. A. C = 25 + d + 100 B. C = 125d C. C = 100d + 25 D. C = 25d + 100 Example 6a

A. Write an equation to represent the total cost C for d days of use. RENTAL COSTS A car rental company charges $25 per day plus a $100 deposit. A. Write an equation to represent the total cost C for d days of use. A. C = 25 + d + 100 B. C = 125d C. C = 100d + 25 D. C = 25d + 100 Example 6a

RENTAL COSTS A car rental company charges $25 per day plus a $100 deposit. B. Compare this rental cost to a company which charges a $50 deposit but $35 per day for use. If a person expects to rent a car for 9 days, which company offers the better rate? A. first company B. second company C. neither D. cannot be determined Example 6b

RENTAL COSTS A car rental company charges $25 per day plus a $100 deposit. B. Compare this rental cost to a company which charges a $50 deposit but $35 per day for use. If a person expects to rent a car for 9 days, which company offers the better rate? A. first company B. second company C. neither D. cannot be determined Example 6b