Teacher’s note: Slides 2-4 are optional of course. I thought it was a good problem to quickly review slope. Pacing: 2 days

There has got to be some “measurable” way to get this aircraft to clear such obstacles. Discuss how you might radio a pilot and tell him or her how to adjust the slope of their flight path in order to clear the mountain. If the pilot doesn’t change something, he / she will not make it home for Christmas. Would you agree? Consider the options: 1) Keep the same slope of his / her path. Not a good choice! 2) Go straight up. Not possible! This is an airplane, not a helicopter.

Fortunately, there is a way to measure a proper “slope” to clear the obstacle. We measure the “change in height” required and divide that by the “horizontal change” required.

y x 10000

Slopes of Parallel & Perpendicular Lines Unit 11 Notes

Rules for m of ll and  Lines Slopes of ll lines are the same. ll lines have equal slopes. Any two vertical lines are ll. Slopes of  lines are flipped and reversed. Opposite reciprocals Vertical & horizontal lines are .

If the lines are perpendicular then: m1 m2 = -1 So if the slope of the first line is 3/8, then the slope of the second is ________ Just flip the first slope and change its sign  3/8  -8/3 = -24/24 = -1 ✔

On the corner of your notes write down whether each of the next few pairs of lines are parallel, perpendicular, or neither.

slope of line r = slope of line s = These lines are parallel.

2. slope of line t = slope of line u = These lines are perpendicular. does m1 m2 = -1 ?

3. slope of line t = slope of line u = These lines are parallel. the slopes are both 3/4

4. slope of line t = slope of line u = NEITHER!! Slopes are not congruent or opposite reciprocals of each other. Do the two slopes multiply to make -1?

5. slope of line t = slope of line u = 7 Perpendicular. . 7 = -1

Example 1: Are the following lines parallel? 4y – 12x = 20 y = 3x – 1 Put them both in y-intercept form y = mx + b See if their slopes are the same, if so then they are ll. y = -3x + 1 Slopes are equal, so the lines are ll. -4y – 12x = 20 -4y = 12x + 20 y =-3x + 5

Example 2: Line r contains points P(0,3) & Q(-2,5). Line t contains points R(0,-7) & S(3, -10). Are they ll, , or neither? x1 y1 x2 y2 x1 y1 x2 y2 (0, 3) and (-2, 5) m = (y2 – y1)/(x2 – x1) = (5 – 3)/(-2 – 0) = 2/-2 = -1 (0, -7) and (3, -10) m = (y2 – y1)/(x2 – x1) = (-10 – (-7))/(3 - 0) = -3/3 = -1 Slopes are equal so the lines are ll.

Example 3: Write an equation in point – slope form for the line parallel to 6x – 3y = 9 that contains point (-5, -8). Step 1: Find the slope of the given line. 6x – 3y = 9 -3y = -6x + 9 y = 2x - 3 y – y1 = m(x – x1) y – (-8) = 2(x – (-5)) y + 8 = 2x + 10 y = 2x + 2 Plug 2 in for m in the formula & plug in the point. (-5, -8) x1 y1

Example 4: Write an equation for the line  to 3x + y = -5 that contains the point (-3, 7). Step 2: Plug in your m and the point. (-3, 7) y – y1 = m(x – x1) y – 7 = 1/3(x – (-3)) y – 7 = (1/3)x + 1 y = (1/3)x + 8 Step 1: Find slope of given line then flip and reverse it. 3x + y = -5 y = -3x – 5 m = -3 m = 1/3 x1 y1

You try  Are the following lines ll, , or neither? y = 1/2x + 3

3) Write an equation of the line perpendicular to the graph of YOU Try More  1) Write an equation of the line parallel to the graph of y = 2x – 5 that passes through the point (3, 7). y = 2x + 1 2) Write an equation of the line parallel to the graph of 3x + y = 6 that passes through the point (1, 4). y = -3x + 7 3) Write an equation of the line perpendicular to the graph of that passes through the point ( - 3, 8). y = -4x -4

Assignment: Day 1 Blue workbook: pg. 54 #28-30, pg. 55 #7-18 Day 2 Practice ws “working w/ ll and  lines”