Find and Use Slopes of Lines Warm Up Lesson Presentation Lesson Quiz

Warm-Up 1. Evaluate if a = 5, b = 2, c = 1, and d = 7. a – b c – d ANSWER 12 – 2. Solve . x – 3 3 – 4 = 15 ANSWER 14 5

Warm-Up 3. What is the reciprocal of ? 2 3 ANSWER 32 4. Julie was thinking of a number. The product of her number and 6 is –1. What was Julie’s number? ANSWER 1 6 –

Example 1 Find the slope of line a and line d. SOLUTION y2 – y1 x2 – x1 = = 4 – 2 6 – 8 = 2 – 2 Slope of line a: m = – 1 y2 – y1 x2 – x1 = = 4 – 0 6 – 6 = 4 Slope of line d: m which is undefined.

Guided Practice Use the graph in Example 1. Find the slope of the line. Line b Line c 2 ANSWER ANSWER

Example 2 Find the slope of each line. Which lines are parallel? SOLUTION Find the slope of k1 through (– 2, 4) and (– 3, 0). m1 = 0 – 4 – 3 – (– 2 ) = – 4 – 1 = 4 Find the slope of k2 through (4, 5) and (1, 3). m2 1 – 5 3 – 4 = = – 4 – 1 = 4

Example 2 Find the slope of k3 through (6, 3) and (5, – 2). m3 – 2 – 3 5 – 6 = = – 5 – 1 5 Compare the slopes. Because k1 and k2 have the same slope, they are parallel. The slope of k3 is different, so k3 is not parallel to the other lines.

Guided Practice Line m passes through (–1, 3) and (4, 1). Line t passes through (–2, –1) and (3, – 3). Are the two lines parallel? Explain how you know. Yes; they have the same slope. ANSWER

Example 3 Line h passes through (3, 0) and (7, 6). Graph the line perpendicular to h that passes through the point (2, 5). SOLUTION STEP 1 Find the slope m1 of line h through (3, 0) and (7, 6). m1 = 6 – 0 7 – 3 = 6 4 = 3 2

Use the rise and run to graph the line. Example 3 STEP 2 Find the slope m2 of a line perpendicular to h. Use the fact that the product of the slopes of two perpendicular lines is –1. m2 = 3 2 – 1 Slopes of perpendicular lines m2 = – 2 3 2 3 Multiply each side by . STEP 3 Use the rise and run to graph the line.

Example 4 Quadrilateral ABCD has vertices A(7, 3), B(5, 6), C(–1, 2), and D(1, –1). Find the slope of the sides and the length of the sides. Then tell whether quadrilateral ABCD is a rectangle, a square, or neither. Explain your reasoning. SOLUTION Find slopes. slope of AB slope of BC slope of CD slope of DA

Example 4 Find lengths. AB BC CD DA Quadrilateral ABCD is a rectangle, but it is not a square. The products of the slopes of consecutive sides is , so ABCD has four right angles. The four sides are not all the same length.

Guided Practice Line n passes through (0, 2) and (6, 5). Line m passes through (2, 4) and (4, 0). Is n m? Explain. ANSWER Yes; the product of their slopes is – 1. In Example 4, show that the diagonals of ABCD are congruent. ANSWER AC = BD = , so AC BD.

Example 5 Roller Coasters During the climb on the Magnum XL-200 roller coaster, you move 41 feet upward for every 80 feet you move horizontally. At the crest of the hill, you have moved 400 feet forward. a. Making a Table: Make a table showing the height of the Magnum at every 80 feet it moves horizontally. How high is the roller coaster at the top of its climb?

Example 5 b. Calculating : Write a fraction that represents the height the Magnum climbs for each foot it moves horizontally. What does the numerator represent? c. Using a Graph: Another roller coaster, the Millennium Force, climbs at a slope of 1. At its crest, the horizontal distance from the starting point is 310 feet. Compare this climb to that of the Magnum. Which climb is steeper?

The Magnum XL-200 is 205 feet high at the top of its climb. Example 5 SOLUTION a. The Magnum XL-200 is 205 feet high at the top of its climb. = rise run 41 80 = = 41 80 80 80 = 0.5125 1 Slope of the Magnum b. The numerator, 0.5125, represents the slope in decimal form.

Example 5 Use a graph to compare the climbs. Let x be the horizontal distance and let y be the height. Because the slope of the Millennium Force is 1, the rise is equal to the run. So the highest point must be at (310, 310). c. The graph shows that the Millennium Force has a steeper climb, because the slope of its line is greater (1 > 0.5125). ANSWER

Guided Practice Line q passes through the points (0, 0) and (– 4, 5). Line t passes through the points (0, 0) and (–10, 7). Which line is steeper, q or t ? 6. Line q ANSWER 7. What If? Suppose a roller coaster climbed 300 feet upward for every 350 feet it moved horizontally. Is it more steep or less steep than the Magnum? than the Millennium Force? more steep than the Magnum; less steep than the Millennium Force ANSWER

Lesson Quiz 1. Find the slope of the line containing the points (4, – 3) and (5, 2). 5 ANSWER 2. Line k passes through the points (– 1, 2) and (3, 5). Line n passes through the points (3, 7) and (6, 3). Are lines k and n parallel, perpendicular, or neither? Perpendicular ANSWER 3. A highway has a grade of 7 percent. For each 200 feet it goes horizontally, how many feet does it rise? 14 ft ANSWER