Perpendicular Lines 1

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Geometry

Analytic Geometry

Perpendicular Lines

Perpendicular Lines 1

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Question 1 of 8

1. Question
Are the two lines at right angles with each other?
- 1. Yes
- 2. No
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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 y-intercept (where the line cuts the y-axis)

The gradient is given by the coefficient of $x$ or the value of $m$ .

$y$ $=$ $-1/2x+4$

$m$ $=$ $-1/2$

$y$ $=$ $2x-3$

$m$ $=$ $2$

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

>Find the equation of a line that passes through $(3,2)$ and is perpendicular to $y=x+5$

1. $y=-x+5$
2. $y=-3x+5$
3. $y=x+15$
4. $y=2x+3$

Question 3 of 8

>Find the equation of a line that passes through $(3,7)$ and is perpendicular to $y=3x-2$

1. $y=-x+8/3$
2. $y=-x+3$
3. $y=3x+4$
4. $y=-1/3x+8$

Question 4 of 8

Find the equation of a line that passes through

$(6,4)$ and is perpendicular to

$3x-4y+8=0$

1. $y=-4/3x+12$
2. $y=4/3x+12$
3. $y=3/4x+12$
4. $y=-4/3x-12$

Question 5 of 8

Check if the line passing through

$R(-2,2)$ and

$S(1,5)$ is perpendicular to the line passing through

$T(1,2)$ and

$U(4,-1)$ .

1. Perpendicular
2. Not Perpendicular

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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 formula

Line 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

>Find the equation of a line that passes through $(2,2)$ and is perpendicular to $y=-2x+6$

1. $y=x-1$
2. $y=2x+4$
3. $y=x+2$
4. $y=1/2x+1$

Question 7 of 8

>Find the equation of a line that passes through $(-3,1)$ and is perpendicular to $y=4x+1$

1. $y=-x+1$
2. $y=-1/4x+1/4$
3. $y=-x+1/4$
4. $y=-4x+4$

Question 8 of 8

>Find the equation of a line that passes through $(-6,4)$ and is perpendicular to $3y=2x-3$

1. $y=-3x-5$
2. $y=-3x-10$
3. $y=-3/2x-5$
4. $y=x-5$

$y- color(royalblue)(y_1)$	$=$	$color(tomato)(m)(x- color(royalblue)(x_1))$	Point Slope Formula
$y- color(royalblue)(2)$	$=$	$color(tomato)(-1)(x- color(royalblue)(3))$	Substitute values
$y-2$	$=$	$-x+3$
$y-2 color(crimson)(+2)$	$=$	$-x+3 color(crimson)(+2)$	Add $2$ to both sides
$y$	$=$	$-x+5$	Simplify

$y- color(royalblue)(y_1)$	$=$	$color(tomato)(m)(x- color(royalblue)(x_1))$	Point Slope Formula
$y- color(royalblue)(7)$	$=$	$color(tomato)(-1/3)(x- color(royalblue)(3))$	Substitute values
$y-7$	$=$	$-1/3x+1$
$y-7 color(crimson)(+7)$	$=$	$-1/3x+1 color(crimson)(+7)$	Add $7$ to both sides
$y$	$=$	$-1/3x+8$	Simplify

$3x-4y+8$	$=$	$0$
$3x-4y+8$ $+4y$	$=$	$0$ $+4y$	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

$m_1$	$=$	$3/4$

$y-$ $y_1$	$=$	$m$ $(x-$ $x_1$ $)$	Point Gradient Formula

$y-$ $4$	$=$	$-4/3$ $(x-$ $6$ $)$	Substitute values

$y-4$	$=$	$-4/3x+8$

$y-4$ $+4$	$=$	$-4/3x + 8$ $+4$	Add $4$ to both sides

$y$	$=$	$-4/3x +12$	Simplify

$y- color(royalblue)(y_1)$	$=$	$color(tomato)(m)(x- color(royalblue)(x_1))$	Point Slope Formula
$y- color(royalblue)(2)$	$=$	$color(tomato)(1/2)(x- color(royalblue)(2))$	Substitute values
$y-2$	$=$	$1/2x-1$
$y-2 color(crimson)(+2)$	$=$	$1/2x-1 color(crimson)(+2)$	Add $2$ to both sides
$y$	$=$	$1/2x+1$	Simplify

Perpendicular Lines 1

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Quiz summary

Information

1. Question

Hint

Correct!

Gradient Intercept Form: $y=mx+b$

2. Question

Keep Going!

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

Remember

3. Question

Keep Going!

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

Remember

4. Question

Hint

Excellent!

Point Gradient Form: $y-$ $y_1$ $=$ $m$ $(x-$ $x_1$ $)$

Remember

5. Question

Hint

Great Work!

Gradient Formula

Remember

6. Question

Keep Going!

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

Remember

7. Question

Keep Going!

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

Remember

8. Question

Keep Going!

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

Remember

Quizzes

$m_1$	$=$	$1$

	$=$	$1/1$	Flip the number upside down
$m_2$	$=$	$-1$	Change the sign

$m_1$	$=$	$3$

	$=$	$1/3$	Flip the number upside down
$m_2$	$=$	$-1/3$	Change the sign

$m_1$	$=$	$3/4$

	$=$	$-4/3$	Flip the number upside down

$m_2$	$=$	$-4/3$	Change the sign

$m_1$	$=$	$-2$

	$=$	$-1/2$	Flip the number upside down
$m_2$	$=$	$1/2$	Change the sign

$m_1$	$=$	$4$

	$=$	$1/4$	Flip the number upside down
$m_2$	$=$	$-1/4$	Change the sign

$3y$	$=$	$2x-3$
$y$	$=$	$2/3x-3/3$	Divide all terms by $3$

$y$	$=$	$2/3x-1$

$m_1$	$=$	$2/3$

	$=$	$3/2$	Flip the number upside down
$m_2$	$=$	$-3/2$	Change the sign

Perpendicular Lines 1

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Quiz summary

Information

1. Question

Hint

Correct!

Gradient Intercept Form: y=mx+by=mx+b

2. Question

Keep Going!

Point Slope Form: y-y1=m(x-x1)y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))

Remember

3. Question

Keep Going!

Point Slope Form: y-y1=m(x-x1)y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))

Remember

4. Question

Hint

Excellent!

Point Gradient Form: y-y-y1y_1==mm(x-(x-x1x_1))

Remember

5. Question

Hint

Great Work!

Gradient Formula

Remember

6. Question

Keep Going!

Point Slope Form: y-y1=m(x-x1)y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))

Remember

7. Question

Keep Going!

Point Slope Form: y-y1=m(x-x1)y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))

Remember

8. Question

Keep Going!

Point Slope Form: y-y1=m(x-x1)y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))

Remember

Quizzes

Gradient Intercept Form: $y=mx+b$

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

Point Gradient Form: $y-$ $y_1$ $=$ $m$ $(x-$ $x_1$ $)$

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$

Point Slope Form: $y- color(royalblue)(y_1)= color(tomato)(m)(x- color(royalblue)(x_1))$