U2B L3 Parallel and Perpendicular Lines Skill Builder
Completion requirements
Unit 2B
Derivatives Part 2
Lesson 3: Finding Tangent Lines Along Curves
Skill Builder
Slope and Parallel Lines
Parallel lines have a constant separation and never intersect.
Parallel lines have the same slope, but different intercepts.
| The slopes of vertical lines are not defined, so they are a special case of parallel lines. Vertical lines are parallel to each other. |
Determine an equation for a line that is parallel to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»3«/mn»«mi»x«/mi»«mo»+«/mo»«mn»9«/mn»«/mrow»«/mstyle»«/math» and passes
through the point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»2«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»6«/mn»«/mrow»«/mfenced»«/mstyle»«/math».
The slope of the original line is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math», so the parallel line must also have a slope of «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»3«/mn»«/mstyle»«/math». Knowing both the slope and a point on the line is enough information to write an equation for the new line, using the slope-point form.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«mo»-«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»m«/mi»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«mo»-«/mo»«mfenced»«mrow»«mo»-«/mo»«mn»6«/mn»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«mo»+«/mo»«mn»6«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»3«/mn»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfenced»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Slope and Perpendicular Lines
Perpendicular lines meet at right angles.
| Recall that the reciprocal of a number can be found by swapping the numerator and denominator. A number that isn’t expressed as a fraction has a denominator of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mn»1«/mn»«/mstyle»«/math».
For example, the number «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» can also be expressed as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mn»2«/mn»«mn»1«/mn»«/mfrac»«/mstyle»«/math». Therefore, the reciprocal of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mstyle»«/math» and the negative reciprocal of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/mrow»«/mstyle»«/math». |
For the line «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mrow»«mi»y«/mi»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»=«/mo»«mo»-«/mo»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»5«/mn»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math»,
determine the slope of a line
a.
parallel to the given line.
b.
perpendicular to the given line.
a.
The slope of a given line is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«/mrow»«/mstyle»«/math». The slope of the parallel
line is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«/mrow»«/mstyle»«/math».
b.
The slope of the given line is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mfrac»«mn»3«/mn»«mn»4«/mn»«/mfrac»«/mrow»«/mstyle»«/math». The negative reciprocal
is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mfenced»«mrow»«mo»-«/mo»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/mfenced»«mo»=«/mo»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/mstyle»«/math»,
so the slope of the new line is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mn»4«/mn»«mn»3«/mn»«/mfrac»«/mstyle»«/math».