Parallel Lines Topic 4.4.1

Parallel Lines 4.4.1 Topic California Standard: What it means for you: 8.0 Students understand the concepts of parallel lines and perpendicular lines and how their slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. What it means for you: You’ll work out the slopes of parallel lines and you’ll test if two lines are parallel. Key words: parallel intersect

Topic 4.4.1 Parallel Lines Now that you’ve practiced finding the slope of a line, you can use the method on a special case — parallel lines. Remember the rise over run formula from Topic 4.3.1: Slope = , provided Dx ¹ 0 vertical change horizontal change = rise run Dy Dx You can use it to prove that two lines are parallel.

Parallel Lines 4.4.1 Topic Parallel Lines Never Meet y-axis x-axis Parallel lines are two or more lines in a plane that never intersect (cross). These lines are all parallel. No matter how long you draw them, they’ll never meet. The symbol || is used to indicate parallel lines — you read this symbol as “is parallel to.” So, if l1 and l2 are lines, then l1 || l2 means “line l1 is parallel to line l2.”

Parallel Lines 4.4.1 Topic Parallel Lines Have Identical Slopes You can determine whether lines are parallel by looking at their slopes. Two lines are parallel if their slopes are equal.

Topic 4.4.1 Parallel Lines Example 1 Prove that the three lines A, B, and C shown on the graph are parallel. y-axis x-axis Solution Using the rise over run formula: slope = , you can see that they all have a slope of = . 2 4 1 Dy Dx 4 2 4 2 A 4 2 B C Solution follows…

Parallel Lines 4.4.1 Topic Guided Practice 1. Two lines on the same plane that never intersect are called ……………… lines. 2. To determine if two lines are parallel you can look at their ……………… 3. Prove that the line f defined by y – 3 = (x – 4) is parallel to line g defined by y – 6 = (x + 1). 2 3 parallel slopes Lines f and g are both written in point-slope form, so both lines have slope . The slopes are the same, so the lines are parallel. 2 3 Solution follows…

Parallel Lines 4.4.1 Topic Vertical Lines Don’t Have Defined Slopes Vertical lines are parallel, but you can’t include them in the definition on slide 5 because their slopes are undefined. Points on a vertical line all have the same x-coordinate, so they are of the form (c, y1) and (c, y2). The slope of a vertical line is undefined because m = = is not defined. y2 – y1 c – c

Topic 4.4.1 Parallel Lines Test if Lines are Parallel by Finding Slopes To check if a pair of lines are parallel, just find the slope of each line. If the slopes are equal, the lines are parallel. Remember — if you’re given two points on a line, you can find the slope of the line using the formula: y2 – y1 x2 – x1 m =

Topic 4.4.1 Parallel Lines Example 2 Show that the straight line through (2, –3) and (–5, 1) is parallel to the straight line joining (7, –1) and (0, 3). Solution Step 1: Find the slope of each line using m = y2 – y1 x2 – x1 1 – (–3) –5 – 2 m1 = = = – 4 –7 7 3 – (–1) 0 – 7 m2 = = = – 4 –7 7 Step 2: Compare the slopes and draw a conclusion. – = – , so m1 = m2. 4 7 So the straight line through (2, –3) and (–5, 1) is parallel to the straight line through (7, –1) and (0, 3). Solution follows…

Parallel Lines 4.4.1 Topic Guided Practice 4. Show that line a, which goes through points (7, 2) and (3, 3), is parallel to line b joining points (–8, –4) and (–4, –5). Both lines have the same slope, so they are parallel. mb = = = – –5 – (–4) –4 – (–8) –1 4 ma = = = – 3 – 2 3 – 7 1 –4 5. Show that the line through points (4, 3) and (–1, 3) is parallel to the line though points (–6, –1) and (–8, –1). Both lines have slopes of zero, so they are parallel. m1 = = = 0 3 – 3 –1 – 4 –5 m2 = = = 0 –1 – (–1) –8 – (–6) –2 Solution follows…

Parallel Lines 4.4.1 Topic Guided Practice 6. Determine if line f joining points (1, 4) and (6, 2) is parallel to line g joining points (0, 8) and (10, 4). Lines f and g both have slopes of – , so the lines are parallel. 2 5 7. Determine if the line through points (–5, 2) and (3, 7) is parallel to the line through points (–5, 1) and (–3, 6). Line 1 has slope and line 2 has slope , so the lines are not parallel. 5 8 2 8. Determine if the line through points (–8, 4) and (–8, 3) is parallel to the line through points (6, 3) and (–4, 3). Line 1 has an undefined slope (it’s vertical) and line 2 has slope 0 (it’s horizontal). So the lines are not parallel. Solution follows…

Topic 4.4.1 Parallel Lines Example 3 Find the equation of a line through (–1, 4) that is parallel to the straight line joining (5, 7) and (–6, –8). Solution This parallel line problem seems tougher but it’s not too hard. Step 1: Find the slope of the line through (5, 7) and (–6, –8). –8 – 7 –6 – 5 m1 = = = – –15 –11 15 11 Step 2: The slope of the line through (–1, 4) must be equal to m1 since the lines are parallel. So, m1 = m2 = – 15 11 Solution continues… Solution follows…

Topic 4.4.1 Parallel Lines Example 3 Find the equation of a line through (–1, 4) that is parallel to the straight line joining (5, 7) and (–6, –8). Solution (continued) Step 3: Now use the point-slope formula to find the equation of the line through point (–1, 4) with slope . 15 11 y – y1 = m(x – x1) 15 11 Þ y – 4 = [x – (–1)] Þ 11y – 44 = 15(x + 1) Þ 11y – 44 = 15x + 15 Þ Equation: 11y – 15x = 59

Parallel Lines 4.4.1 Topic Guided Practice 9. Find the equation of the line through (–3, 7) that is parallel to the line joining points (4, 5) and (–2, –8). 10. Find the equation of the line through (6, –4) that is parallel to the line joining points (–1, 6) and (7, 3). 11. Find the equation of the line through (–1, 7) that is parallel to the line joining points (4, –3) and (8, 6). m = = = –8 – 5 –2 – 4 –13 –6 13 6 y – 7 = [x – (–3)] Þ 6y – 42 = 13x + 39 Þ 6y – 13x = 81 13 6 m = = = – 3 – 6 7 – (–1) –3 8 3 y – (–4) = – [x – 6] Þ 8y + 32 = –3x + 18 Þ 8y + 3x = –14 3 8 m = = 6 – (–3) 8 – 4 9 4 y – 7 = [x – (–1)] Þ 4y – 28 = 9x + 9 Þ 4y – 9x = 37 9 4 Solution follows…

Parallel Lines 4.4.1 Topic Guided Practice 12. Write the equation of the line through (–3, 5) that is parallel to the line joining points (–1, 2.5) and (0.5, 1). 13. Write the equation of the line through (–2, –1) that is parallel to the line x + 3y = 6. m = = = – 1 1 – 2.5 0.5 – (–1) –1.5 1.5 y – 5 = –1[x – (–3)] Þ y – 5 = –x – 3 Þ y + x = 2 Find two points on the line x + 3y = 6. For example, (0, 2) and (6, 0). Use these points to find m. m = = – 0 – 2 6 – 0 1 3 y – (–1) = – [x – (–2)] Þ 3y + 3 = –x – 2 Þ 3y + x = –5 1 3 Solution follows…

Parallel Lines 4.4.1 Topic Independent Practice 1. Line l1 has slope and line l2 has slope . What can you conclude about l1 and l2? 1 2 1 2 The lines are parallel. 2. Line l1 has a slope of – . If l1 || l2, then what is the slope of l2? 1 3 1 3 – 3. Show that all horizontal lines are parallel. Points on a horizontal line all have the same y-coordinate, c, where c is a constant. So, for any two points, y2 – y1 = c – c = 0. This means horizontal lines all have a slope of zero, so they are all parallel to each other. Solution follows…

Parallel Lines 4.4.1 Topic Independent Practice 4. Show that the line through the points (5, –3) and (–8, 1) is parallel to the line through (13, –7) and (–13, 1). m1 = m2, so the lines are parallel. m1 = = – 1 – (–3) –8 – 5 4 13 m2 = = – = – 1 – (–7) –13 – 13 8 26 5. Determine if the line through the points (5, 4) and (0, 9) is parallel to the line through (–1, 8) and (4, 0). Line 1 has slope –1 and line 2 has slope – , so the lines are not parallel. 8 5 6. Determine if the line through the points (–2, 5) and (6, 0) is parallel to the line through (8, –1) and (0, 4). Both lines have slopes of – , so the lines are parallel. 5 8 Solution follows…

Parallel Lines 4.4.1 Topic Independent Practice 7. Determine if the line through the points (4, –7) and (4, –4) is parallel to the line through (–5, 1) and (–5, 5). The lines are parallel since they are both vertical lines. 8. Determine if the line through the points (–2, 3) and (–2, –2) is parallel to the line through (1, 7) and (–6, 7). The lines are not parallel, since one is vertical and the other is horizontal. 9. Find the equation of the line through (1, –2) that is parallel to the line joining the points (–3, –1) and (8, 7). 11y + 8x = –30 10. Find the equation of the line through (–5, 3) that is parallel to the line joining the points (–2, 6) and (8, –1). 10y + 7x = –5 11. Write the equation of the line through (0, 6) that is parallel to the line 3x + 2y = 6. 2y + 3x = 12 Solution follows…

Parallel Lines 4.4.1 Topic Round Up When you draw lines with different slopes on a set of axes, you might not see where they cross. But remember, you are only looking at a tiny bit of the lines — they go on indefinitely in both directions. If they don’t have identical slopes, they’ll cross sooner or later.