SLOPE AND PARALLEL AND PERPENDICULAR LINES

You will need the worksheet that goes along with this power point presentation, a pencil, a ruler, and a piece of graph paper. Read and follow directions carefully. At the end of The presentation, you should know something about the slopes of parallel lines and the slopes of perpendicular lines.

REVIEW: Parallel lines (//) are lines that never intersect. Perpendicular lines () are lines that intersect to form right angles.

SLOPE One nice thing about slope is that it’s a concept you already know about from everyday life. People talk about the slope of hills; golf commentators discuss the gentle slope of putting greens. So experience tells you that slope has to do with steepness. In math, slope has a specific, technical meaning: It means the steepness of lines on the coordinate plane. From life, you also have a sense of what steepness means. For example, after climbing two hills, you should be able to say which one was steeper (at least your leg muscles will tell you the next day). But math, exact science that it is, allows you to measure steepness with perfect, numerical precision.

In algebra, the symbol for slope is the letter m. Here are some other things to know about slope: • A horizontal line, having no steepness, has a slope of 0. • A vertical line, having absolute steepness, has a slope that is undefined. • Lines that rise from left to right have a positive slope. • Lines that fall from left to right have a negative slope.

What is the formuula for the slope of a line? OR The formula for the slope of a line is: rise y2 - y1 OR m = m = run x2 - x1 where (x1, y1) and (x2, y2) are the coordinates of any two points on the line. (x2, y2) (x1, y1)

Find the slope of the lines passing through these points. Show your work and write your answers on your piece of paper (#6). a) (4, 3) and (9, 6) b) (8, -2) and (6, 5) - 3 9 - 4 5 - -2 6 - 8 = 3/5 = -7/2 c) (-10, 5) and (15, -20) d) (-3, -6) and (8, 7)

Count the slope when you are given a line on a graph. To get from one point to the other on the line, you have to go up 2 and over 3 to the right, so the slope is ⅔. Over 3 Up 2

Find the slope of the line graphed below. Notice that to get from one point to the next on this graph you have to go down 2 and to the right one, so the slope is -2/1.

If you are given a point on a line and the slope of the line, you can graph the line. • Graph the point that you are given. This will be the starting point. • Start at the point you graphed and count the slope. Graph the line through (-6, -5) with slope 3/2.

The line through (-4, 2) with slope of 4/5. Use the graph by #7 on your worksheet to complete the following graphs: The line through (-4, 2) with slope of 4/5. b) The line through (1, 2) with slope of -3/4. c) The line through (3, -2) with slope of 3.

On your graph for #8, graph the line through (2, 3) with a slope of ¾. On the same graph, graph the equation y = ¾ x + -4. Click here if you need to review how to graph a linear equation. What do you notice about the two lines that you graphed? What do you notice about the slopes of the two lines?

On the graph for #9, graph AB and CD. Click here if you forgot how to find slope. Find the slope of each line. The slope of AB = 6/5 5/6 -5/6 -6/5 The slope of CD = 6/5 5/6 -5/6 -6/5 Click on the correct answer. What do you notice about the two lines that you graphed? What do you notice about the slopes of the two lines?

What can you conclude about the slopes of parallel lines? The slopes of parallel lines are equal or the same!! The slope of a line is 1/3. What is the slope of a line parallel to this one? 3 -1/3 1/4 1/3

Line b has a slope of 2/3 and line d has a slope of 1/3. Is b // d? yes no E (2, -11), F (4, -5), G (2, 8) and H (5, 17) Without graphing, determine if EF// GH. Show all work on your paper next to #10. (think slope) Click here if you forgot how to find slope.

Graph the line through (-5, -8) with a slope of ½. (Do this on the graph for #11). On the same graph, graph the line that passes through (-9, 7) with a slope of -2. What do you notice about the two lines? Is there a relationship between the slopes?

Graph the two equations below on the graph for #12. y = 2/3 x + 2 Click here if you need help graphing linear equations. y = -3/2 x + -3 What do you notice about the two lines? What do you notice about the slopes? (Hint: try multiplying them)

What do you notice about the two lines? L (-4, 0) M (0, 3) N (1, 7) P (4, 3) Graph LM and NP on #13. What do you notice about the two lines? Click on the slope of LM : ¾ 4/3 -3/4 -4/3 Click on the slope of NP : ¾ 4/3 -3/4 -4/3 What do you notice about the slopes of these two perpendicular lines? Hint: try multiplying them again. Click here if you forgot how to find slope.

What can you conclude about the slopes of perpendicular lines? The product of the slopes of perpendicular lines is -1. Notice that if the slope of the original line is ¾, to find the slope of the perpendicular line (-4/3) you flip the original slope (reciprocal) and make it the opposite. The slope of a line is -5/7. What is the slope of the perpendicular line? (flip and opposite) 5/7 7/5 -5/7 -7/5

Line a has a slope of -1/2 and line c has a slope of 2. Is a  b? yes no Q (-3, -2) R (9, 1) S (3, 6) T (5, -2) Without graphing, determine if QR  ST. Think slope. Show all work on your paper next to #14. Click here if you forgot how to find slope!

Slopes of parallel and perpendicular lines The equation for a line is y = ¼ x + 5. What is the slope of a line parallel to this one? 5 ¼ 4/1 -1/4 -1/5 -4/1 What is the slope of a line perpendicular to this one? 5 ¼ 4/1 -1/4 -1/5 -4/1

Complete #15 & #16 on your worksheet. Did you complete #15 & #16?

Okay, I get it! The slopes of parallel lines are equal! And the slopes of perpendicular lines are opposite reciprocals (their product is -1)! For perpendicular lines I just flip the slope and change to its opposite. That’s what I said!

Complete #17 & #18 on the worksheet and then you are finished!

CORRECT!

Try again.

Review - graphing a linear equation: To graph the equation y = ½ x + 2: 1. Graph the y-intercept (the point where the line crosses the y-axis) - (0,2) Count the slope - rise over run (½) Draw a line through the points Return to previous slide

Try Again. Remember that the slopes of parallel lines are equal.

Try Again. Remember that the slopes of perpendicular lines have a product of -1. So, take the original slope and flip it. Then make it the opposite.

Return to previous slide. What is the formuula for the slope of a line? The formula for the slope of a line is: rise y2 - y1 OR m = m = run x2 - x1 where (x1, y1) and (x2, y2) are the coordinates of any two points on the line. (x2, y2) (x1, y1) Return to previous slide.

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