Perpendicular Lines 1
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Question 1 of 8
1. Question
Are the two lines at right angles with each other?Hint
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Perpendicular lines have gradients which are negative reciprocals of one another.Gradient Intercept Form: `y=mx+b`
 `m` is the gradient of the line
 `b` is the yintercept (where the line cuts the yaxis)
The gradient is given by the coefficient of `x` or the value of `m`.`y` `=` `1/2x+4` `m` `=` `1/2` `y` `=` `2x3` `m` `=` `2` The gradients are negative reciprocals of each other, so the lines are perpendicular or are at right angles with each other.The lines are perpendicular. 
Question 2 of 8
2. Question
>Find the equation of a line that passes through `(3,2)` and is perpendicular to `y=x+5`
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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)(1)x+5` `m_1` `=` `1` Get the negative reciprocal of the `m_1` by flipping it upside down and changing the sign.`m_1` `=` `1` `=` `1/1` Flip the number upside down `m_2` `=` `1` Change the sign Use the point slope formula to find the equation.Point: `(x_1,y_1)=(3,2)`Slope: `m_2=1``y color(royalblue)(y_1)` `=` `color(tomato)(m)(x color(royalblue)(x_1))` Point Slope Formula `y color(royalblue)(2)` `=` `color(tomato)(1)(x color(royalblue)(3))` Substitute values `y2` `=` `x+3` `y2 color(crimson)(+2)` `=` `x+3 color(crimson)(+2)` Add `2` to both sides `y` `=` `x+5` Simplify `y=x+5` 
Question 3 of 8
3. Question
>Find the equation of a line that passes through `(3,7)` and is perpendicular to `y=3x2`
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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)(3)x+2` `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)=(3,7)`Slope: `m_2=1/3``y color(royalblue)(y_1)` `=` `color(tomato)(m)(x color(royalblue)(x_1))` Point Slope Formula `y color(royalblue)(7)` `=` `color(tomato)(1/3)(x color(royalblue)(3))` Substitute values `y7` `=` `1/3x+1` `y7 color(crimson)(+7)` `=` `1/3x+1 color(crimson)(+7)` Add `7` to both sides `y` `=` `1/3x+8` Simplify `y=1/3x+8` 
Question 4 of 8
4. Question
Find the equation of a line that passes through `(6,4)` and is perpendicular to `3x4y+8=0`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, convert the equation into gradientintercept form and identify the gradient.In gradientintercept form `(y=``m``x+b)`, `m` is the gradient.`3x4y+8` `=` `0` `3x4y+8` `+4y` `=` `0` `+4y` Add `4y` on both sides `3x+8` `=` `4y` `3/4x+8/4` `=` `4/4y` Divide all terms by `4` `3/4 x + 2` `=` `y` `y` `=` `3/4``x+2` Identify the slope `m_1` `=` `3/4` Get the negative reciprocal of the `m_1` by flipping it upside down and changing the sign.`m_1` `=` `3/4` `=` `4/3` Flip the number upside down `m_2` `=` `4/3` Change the sign Use the Point Gradient Formula to find the equation.Point: `(x_1,y_1)=(6,4)`Gradient: `m_2=2``y``y_1` `=` `m``(x``x_1``)` Point Gradient Formula `y``4` `=` `4/3``(x``6``)` Substitute values `y4` `=` `4/3x+8` `y4` `+4` `=` `4/3x + 8` `+4` Add `4` to both sides `y` `=` `4/3x +12` Simplify `y=4/3x+12` 
Question 5 of 8
5. Question
Check if the line passing through `R(2,2)` and `S(1,5)` is perpendicular to the line passing through `T(1,2)` and `U(4,1)`.Hint
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Gradient Formula
$$m=\frac{\color{#9a00c7}{y_2}\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}\color{#00880a}{x_1}}$$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`(``x_1``,``y_1``)` `=` `S(``2``,``2``)` `(``x_2``,``y_2``)` `=` `R(``1``,``5``)` `m_1` `=` $$\frac{\color{#9a00c7}{y_2}\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}\color{#00880a}{x_1}}$$ Gradient Formula `=` $$\frac{\color{#9a00c7}{5}\color{#9a00c7}{2}}{\color{#00880a}{1}(\color{#00880a}{2})}$$ Substitute values `=` `3/3` Simplify `=` `1` Line 2`(``x_1``,``y_1``)` `=` `T(``1``,``2``)` `(``x_2``,``y_2``)` `=` `U(``4``,``1``)` `m_2` `=` $$\frac{\color{#9a00c7}{y_2}\color{#9a00c7}{y_1}}{\color{#00880a}{x_2}\color{#00880a}{x_1}}$$ Gradient Formula `=` $$\frac{\color{#9a00c7}{1}\color{#9a00c7}{2}}{\color{#00880a}{4}\color{#00880a}{1}}$$ Substitute values `=` `3/3` Simplify `=` `1` 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` `=` `1 times 1` `=` `1` Therefore, Line `1` and Line `2` are perpendicular.Perpendicular 
Question 6 of 8
6. Question
>Find the equation of a line that passes through `(2,2)` and is perpendicular to `y=2x+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)(2)x+6` `m_1` `=` `2` Get the negative reciprocal of the `m_1` by flipping it upside down and changing the sign.`m_1` `=` `2` `=` `1/2` Flip the number upside down `m_2` `=` `1/2` Change the sign Use the point slope formula to find the equation.Point: `(x_1,y_1)=(2,2)`Slope: `m_2=1/2``y color(royalblue)(y_1)` `=` `color(tomato)(m)(x color(royalblue)(x_1))` Point Slope Formula `y color(royalblue)(2)` `=` `color(tomato)(1/2)(x color(royalblue)(2))` Substitute values `y2` `=` `1/2x1` `y2 color(crimson)(+2)` `=` `1/2x1 color(crimson)(+2)` Add `2` to both sides `y` `=` `1/2x+1` Simplify `y=1/2x+1` 
Question 7 of 8
7. Question
>Find the equation of a line that passes through `(3,1)` and is perpendicular to `y=4x+1`
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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+1` `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)=(3,1)`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)(3))` Substitute values `y1` `=` `1/4x3/4` `y1 color(crimson)(+1)` `=` `1/4x3/4 color(crimson)(+1)` Add `1` to both sides `y` `=` `1/4x+1/4` Simplify `y=1/4x+1/4` 
Question 8 of 8
8. Question
>Find the equation of a line that passes through `(6,4)` and is perpendicular to `3y=2x3`
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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, convert the equation into gradientintercept form and identify the gradient.In gradientintercept form `(y=``m``x+b)`, `m` is the gradient.`3y` `=` `2x3` `y` `=` `2/3x3/3` Divide all terms by `3` `y` `=` `2/3x1` 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)(2/3)x1` `m_1` `=` `2/3` Get the negative reciprocal of the `m_1` by flipping it upside down and changing the sign.`m_1` `=` `2/3` `=` `3/2` Flip the number upside down `m_2` `=` `3/2` Change the sign Use the point slope formula to find the equation.Point: `(x_1,y_1)=(6,4)`Slope: `m_2=3/2``y color(royalblue)(y_1)` `=` `color(tomato)(m)(x color(royalblue)(x_1))` Point Slope Formula `y color(royalblue)(4)` `=` `color(tomato)(3/2)(x color(royalblue)(6))` Substitute values `y4` `=` `3/2x9` `y4 color(crimson)(+4)` `=` `3/2x9 color(crimson)(+4)` Add `4` to both sides `y` `=` `3/2x5` Simplify `y=3/2x5`
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
 Slope of a Line 1
 Slope of a Line 2
 Slope Intercept Form: Graph an Equation 1
 Slope Intercept Form: Graph an Equation 2
 Slope Intercept Form: Write an Equation 1
 Graph Linear Inequalities 1
 Convert Standard Form and Slope Intercept Form 1
 Convert Standard Form and Slope Intercept Form 2
 Point Slope Form 1
 Point Slope Form 2
 Parallel Lines 1
 Parallel Lines 2
 Perpendicular Lines 1
 Perpendicular Lines 2