Perpendicular Lines 2
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Question 1 of 7
1. Question
Find the equation of a line that passes through `(4,2)` and is perpendicular to `y=-1/2x+9`Hint
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Point Gradient Form: `y-``y_1``=``m``(x-``x_1``)`
- `m` is the gradient of the line
- `(x_1,y_1)` is a point that lies on the line
Remember
The gradients of perpendicular lines are negative reciprocals of each other.First, identify the gradient of the given equation.In gradient-intercept form `(y=``m``x+b)`, `m` is the gradient.`y` `=` `-1/2``x+9` `m_1` `=` `-1/2` Get the negative reciprocal of the `m_1` by flipping it upside down and changing the sign.`m_1` `=` `-1/2` `=` `-2` Flip the number upside down `m_2` `=` `2` Change the sign Use the Point Gradient Formula to find the equation.Point: `(x_1,y_1)=(4,2)`Gradient: `m_2=2``y-``y_1` `=` `m``(x-``x_1``)` Point Gradient Formula `y-``2` `=` `2``(x-``4``)` Substitute values `y-2` `=` `2x-8` `y-2` `+2` `=` `2x-8` `+2` Add `2` to both sides `y` `=` `2x-6` Simplify `y=2x-6` -
Question 2 of 7
2. Question
Check if the line `y=2x+5` is perpendicular to the line `x+2y-6=0`.Hint
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Gradient Intercept Form: `y=``m``x+``b`
- `m` is the gradient of the line
- `b` is the y-intercept (where the line cuts the y-axis)
Remember
To prove the perpendicularity of two lines, the product of their gradients should be equal to `-1`First, solve for the gradient of each line using the gradient formulaLine 1`y` `=` `2``x+5` `m_1` `=` `2` Line 2`x+2y-6` `=` `0` `x+2y-6` `-x+6` `=` `0` `-x+6` Add `-x+6` to both sides `2y` `=` `-x+6` Simplify `2y``-:2` `=` `(-x+6)``-:2` Divide both sides by `2` `y` `=` `-1/2``x + 3` `m_2` `=` `-1/2` To prove that these two lines are perpendicular, check if the product of the two gradients is equal to `-1`.`m_1 times m_2` `=` `2 times-1/2` `=` `-1` Therefore, Line `1` and Line `2` are perpendicular.Perpendicular
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Question 3 of 7
3. Question
>Find the equation of a line that passes through `(-8,5)` and is perpendicular to `y=-4x+2`
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Point Gradient Form: `y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))`
- `color(tomato)(m)` is the slope of the line
- `(\color(royalblue)(x_1,y_1) )` is a point that lies on the line
Remember
The slope of perpendicular lines are negative reciprocals of each other.First, identify the slope of the given equation.In slope intercept form `(y= color(tomato)(m)x+b)`, `color(tomato)(m)` is the slope.`y` `=` `color(tomato)(-4)x+2` `m_1` `=` `-4` Get the negative reciprocal of the `m_1` by flipping it upside down and changing the sign.`m_1` `=` `-4` `=` `-1/4` Flip the number upside down `m_2` `=` `1/4` Change the sign Use the point slope formula to find the equation.Point: `(x_1,y_1)=(-8,5)`Slope: `m_2=1/4``y- color(royalblue)(y_1)` `=` `color(tomato)(m)(x- color(royalblue)(x_1))` Point Slope Formula `y- color(royalblue)(5)` `=` `color(tomato)(1/4)(x- color(royalblue)(-8))` Substitute values `y-5` `=` `1/4x+2` `y-5 color(crimson)(+5)` `=` `1/4x+2 color(crimson)(+5)` Add `5` to both sides `y` `=` `1/4x+7` Simplify `y=1/4x+7` -
Question 4 of 7
4. Question
>Find the equation of a line that passes through `(-8,10)` and is perpendicular to `y=4x+6`
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Point Slope Form: `y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))`
- `color(tomato)(m)` is the slope of the line
- `(\color(royalblue)(x_1,y_1) )` is a point that lies on the line
Remember
The slope of perpendicular lines are negative reciprocals of each other.First, identify the slope of the given equation.In slope intercept form `(y= color(tomato)(m)x+b)`, `color(tomato)(m)` is the slope.`y` `=` `color(tomato)(4)x+6` `m_1` `=` `4` Get the negative reciprocal of the `m_1` by flipping it upside down and changing the sign.`m_1` `=` `4` `=` `1/4` Flip the number upside down `m_2` `=` `-1/4` Change the sign Use the point slope formula to find the equation.Point: `(x_1,y_1)=(-8,10)`Slope: `m_2=-1/4``y- color(royalblue)(y_1)` `=` `color(tomato)(m)(x- color(royalblue)(x_1))` Point Slope Formula `y- color(royalblue)(10)` `=` `color(tomato)(-1/4)(x- color(royalblue)(-8))` Substitute values `y-10` `=` `-1/4x+2` `y-10 color(crimson)(+10)` `=` `-1/4x+2 color(crimson)(+10)` Add `10` to both sides `y` `=` `-1/4x+12` Simplify `y=-1/4x+12` -
Question 5 of 7
5. Question
>Find the equation of a line that passes through `(-3,5)` and is perpendicular to `y=5x+6`
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Point Slope Form: `y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))`
- `color(tomato)(m)` is the slope of the line
- `(\color(royalblue)(x_1,y_1) )` is a point that lies on the line
Remember
The slope of perpendicular lines are negative reciprocals of each other.First, identify the slope of the given equation.In slope intercept form `(y= color(tomato)(m)x+b)`, `color(tomato)(m)` is the slope.`y` `=` `color(tomato)(5)x+6` `m_1` `=` `5` Get the negative reciprocal of the `m_1` by flipping it upside down and changing the sign.`m_1` `=` `5` `=` `1/5` Flip the number upside down `m_2` `=` `-1/5` Change the sign Use the point slope formula to find the equation.Point: `(x_1,y_1)=(-3,5)`Slope: `m_2=-1/5``y- color(royalblue)(y_1)` `=` `color(tomato)(m)(x- color(royalblue)(x_1))` Point Slope Formula `y- color(royalblue)(5)` `=` `color(tomato)(-1/5)(x- color(royalblue)(-3))` Substitute values `y-5` `=` `-1/5x+3/5` `y-5 color(crimson)(+5)` `=` `-1/5x+3/5 color(crimson)(+5)` Add `5` to both sides `y` `=` `-1/5x+28/5` Simplify `y=-1/5x+28/5` -
Question 6 of 7
6. Question
>Find the equation of a line that passes through `(4,-1)` and is perpendicular to `y=4x+8`
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Point Slope Form: `y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))`
- `color(tomato)(m)` is the slope of the line
- `(\color(royalblue)(x_1,y_1) )` is a point that lies on the line
Remember
The slope of perpendicular lines are negative reciprocals of each other.First, identify the slope of the given equation.In slope intercept form `(y= color(tomato)(m)x+b)`, `color(tomato)(m)` is the slope.`y` `=` `color(tomato)(4)x+8` `m_1` `=` `4` Get the negative reciprocal of the `m_1` by flipping it upside down and changing the sign.`m_1` `=` `4` `=` `1/4` Flip the number upside down `m_2` `=` `-1/4` Change the sign Use the point slope formula to find the equation.Point: `(x_1,y_1)=(4,-1)`Slope: `m_2=-1/4``y- color(royalblue)(y_1)` `=` `color(tomato)(m)(x- color(royalblue)(x_1))` Point Slope Formula `y- color(royalblue)(-1)` `=` `color(tomato)(-1/4)(x- color(royalblue)(4))` Substitute values `y+1` `=` `-1/4x+1` `y+1 color(crimson)(-1)` `=` `-1/4x+1 color(crimson)(-1)` Subtract `1` to both sides `y` `=` `-1/4x` Simplify `y=-1/4x` -
Question 7 of 7
7. Question
>Find the equation of a line that passes through `(-1,-4)` and is perpendicular to `9x+3y=8`
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Incorrect
Point Slope Form: `y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))`
- `color(tomato)(m)` is the slope of the line
- `(\color(royalblue)(x_1,y_1) )` is a point that lies on the line
Remember
The slope of perpendicular lines are negative reciprocals of each other.First, convert the equation into gradient-intercept form and identify the gradient.In gradient-intercept form `(y=``m``x+b)`, `m` is the gradient.`9x+3y` `=` `8` `9x+3y` `-9x` `=` `8` `-9x` Subtract `9x` on both sides `3y` `=` `8-9x` `3/3y` `=` `8/3-9/3x` Divide all terms by `3` `y` `=` `-3x+8/3` Then, identify the slope of the given equation.In slope intercept form `(y= color(tomato)(m)x+b)`, `color(tomato)(m)` is the slope.`y` `=` `color(tomato)(-3)x+8/3` `m_1` `=` `-3` Get the negative reciprocal of the `m_1` by flipping it upside down and changing the sign.`m_1` `=` `-3` `=` `-1/3` Flip the number upside down `m_2` `=` `1/3` Change the sign Use the point slope formula to find the equation.Point: `(x_1,y_1)=(-1,-4)`Slope: `m_2=1/3``y- color(royalblue)(y_1)` `=` `color(tomato)(m)(x- color(royalblue)(x_1))` Point Slope Formula `y- color(royalblue)(-4)` `=` `color(tomato)(1/3)(x- color(royalblue)(-1))` Substitute values `y+4` `=` `1/3x+1/3` `y+4 color(crimson)(-4)` `=` `1/3x+1/3 color(crimson)(-4)` Subtract `4` to both sides `y` `=` `1/3x-11/3` Simplify `y=1/3x-11/3`
Quizzes
- Distance Between Two Points 1
- Distance Between Two Points 2
- Distance Between Two Points 3
- Midpoint of a Line 1
- Midpoint of a Line 2
- Midpoint of a Line 3
- Gradient of a Line 1
- Gradient of a Line 2
- Gradient Intercept Form: Graph an Equation 1
- Gradient Intercept Form: Graph an Equation 2
- Gradient Intercept Form: Write an Equation 1
- Determine if a Point Lies on a Line
- Graph Linear Inequalities 1
- Graph Linear Inequalities 2
- Convert Between General Form and Gradient Intercept Form 1
- Convert Between General Form and Gradient Intercept Form 2
- Point Gradient and Two Point Formula 1
- Point Gradient and Two Point Formula 2
- Parallel Lines 1
- Parallel Lines 2
- Perpendicular Lines 1
- Perpendicular Lines 2