If two non-vertical lines in the same plane have the same slope, then the lines are parallel .
If two non-vertical lines in the same plane are parallel , then they have the same slope.
Two horizontal lines in the same plane are parallel ; two vertical lines in the same plane are parallel .
Parallel Lines: Concept
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SOLUTION EXAMPLE 1 Write an equation of a parallel line Write an equation of the line that passes through (–3, –5) and is parallel to the line y = 3 x – 1. STEP 1 Identify the slope. The graph of the given equation has a slope of 3. So, the parallel line through (–3, –5) has a slope of 3. STEP 2 Find the y - intercept. Use the slope and the given point. y = m x + b Write slope-intercept form .
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EXAMPLE 1 Write an equation of a parallel line y = 3 x + b – 5 = 3 ( –3 ) + b 4 = b Substitute 3 for m . Substitute 3 for x , and 5 for y . Solve for b . STEP 3 Write an equation. Use y = mx + b . y = 3 x + 4 Substitute 3 for m and 4 for b . Using the point (–3, –5)
If two non-vertical lines in the same plane have slopes that are negative reciprocals, then the lines are perpendicular .
If two non-vertical lines in the same plane are perpendicular , then their slopes are negative reciprocals.
A horizontal line and a vertical line in the same plane are perpendicular .
Perpendicular Lines: Concept
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EXAMPLE 2 Determine whether lines are parallel or perpendicular Determine which lines, if any, are parallel or perpendicular. Line a : y = 5 x – 3 Line b : x + 5 y = 2 Line c : –10 y – 2 x = 0 SOLUTION Find the slopes of the lines .
Line a : The equation is in slope-intercept form .
The slope is 5.
Write the equations for lines b and c in slope-intercept form.
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EXAMPLE 2 5 y = – x + 2 – 10 y = 2 x Determine whether lines are parallel or perpendicular Line a : m = 5. Line b : x + 5 y = 2 Line c : – 10 y – 2 x = 0 y = – x 1 5 x y = 2 5 1 5 + –
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EXAMPLE 2 ANSWER Determine whether lines are parallel or perpendicular
Lines b and c have slopes of – , so they are parallel.
1 5
Line a has a slope of 5, the negative reciprocal
of – , so it is perpendicular to lines b and c .
1 5
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GUIDED PRACTICE for Example 2 Determine which lines, if any, are parallel or perpendicular. Line a : 2 x + 6 y = –3 Line b : y = 3 x – 8 Line c : –1.5 y + 4.5 x = 6 ANSWER parallel: b and c ; perpendicular: a and b , a and c
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SOLUTION EXAMPLE 3 Determine whether lines are perpendicular Line a : 12 y = –7 x + 42 Line b : 11 y = 16 x – 52 Find the slopes of the lines . Write the equations in slope-intercept form . The Arizona state flag is shown in a coordinate plane. Lines a and b appear to be perpendicular. Are they ? STATE FLAG
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EXAMPLE 3 Determine whether lines are perpendicular Line a : 12 y = –7 x + 42 Line b : 11 y = 16 x – 52 ANSWER y = – x + 12 42 7 12 11 52 y = x – 16 11 The slope of line a is – . The slope of line b is . The two slopes are not negative reciprocals, so lines a and b are not perpendicular. 7 12 16 11
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SOLUTION EXAMPLE 4 Write an equation of a perpendicular line Write an equation of the line that passes through (4, –5) and is perpendicular to the line y = 2 x + 3. STEP 1 Identify the slope. The graph of the given equation has a slope of 2. Because the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular line through (4, –5) is . 1 2 –
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EXAMPLE 4 STEP 2 Find the y - intercept. Use the slope and the given point. Write slope-intercept form. Solve for b . STEP 3 Write an equation . y = mx + b Write slope-intercept form . Write an equation of a perpendicular line – 5 = – ( 4 ) + b 1 2 Substitute – for m , 4 for x , and – 5 for y . 1 2 y = m x + b – 3 = b y = – x – 3 1 2 Substitute – for m and –3 for b . 1 2
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GUIDED PRACTICE for Examples 3 and 4 3. Is line a perpendicular to line b ? Justify your answer using slopes. Line a : 2 y + x = –12 Line b : 2 y = 3 x – 8 ANSWER 1 No; the slope of line a is – , the slope of line b is . The slopes are not negative reciprocals so the lines are not perpendicular. 2 3 2
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GUIDED PRACTICE for Examples 3 and 4 4. Write an equation of the line that passes through (4, 3) and is perpendicular to the line y = 4 x – 7 . ANSWER y = – x + 4 1 4
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