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Intermediate Geometry : Perpendicular Lines

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Example Question #8 : How To Find Out If Lines Are Perpendicular

Are the lines of the equations

2x+y = 12

and

x+2y = 12

parallel, perpendicular, or neither?

Possible Answers:

Perpendicular

Parallel

Neither

Correct answer:

Neither

Explanation:

Write each equation in the slope-intercept form y = mx+b by solving for ; the -coefficient is the slope of the line.

2x+y = 12

Subtract from both sides:

2x+y -2x = 12 - 2x

y = -2x+ 12

The line of this equation has slope .

x+2y = 12

Subtract from both sides:

x+2y -x = 12 - x

2y = -x+ 12

Multiply both sides by $\frac{1}{2}$

$\frac{1}{2} \cdot 2y =\frac{1}{2} \cdot ( -x+ 12)$

$y = \frac{1}{2} \cdot( - x)+ \frac{1}{2} \cdot 12$

$y = - \frac{1}{2} x+ 6$

The line of this equation has slope $-\frac{1}{2}$ .

Two lines are parallel if and only if they have the same slope; this is not the case. They are perpendicular if and only if the product of their slopes is ; this is not the case, since

$-2 \times\left ( -\frac{1}{2} \right ) = 1$ .

The correct response is that the lines are neither parallel nor perpendicular.

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Example Question #9 : How To Find Out If Lines Are Perpendicular

Are the lines of the equations

3x+2y = 4

and

2y - 3x = 17

parallel, perpendicular, or neither?

Possible Answers:

Parallel

Perpendicular

Neither

Correct answer:

Neither

Explanation:

Write each equation in the slope-intercept form y = mx+b by solving for ; the -coefficient is the slope of the line.

3x+2y = 4

Subtract from both sides:

3x+2y - 3x = 4 - 3x

2y = - 3x+ 4

Multiply both sides by $\frac{1}{2}$ :

$\frac{1}{2} \cdot 2y =\frac{1}{2} \cdot ( - 3x+ 4)$

$y =\frac{1}{2} \cdot ( - 3x) +\frac{1}{2} \cdot 4$

$y =- \frac{3}{2} x + 2$

The slope is the -coefficient $- \frac{3}{2}$

2y - 3x = 17

Add to both sides:

2y - 3x + 3x = 17 + 3x

2y = 3x+ 17

Multiply both sides by $\frac{1}{2}$ :

$\frac{1}{2} \cdot 2y = \frac{1}{2} \cdot (3x+ 17)$

$y= \frac{1}{2} \cdot 3x+ \frac{1}{2} \cdot 17$

$y= \frac{3}{2} x+ \frac{17}{2}$

The slope is the -coefficient $\frac{3}{2}$ .

Two lines are parallel if and only if they have the same slope; this is not the case. They are perpendicular if and only if the product of their slopes is ; this is not the case, since $- \frac{3}{2} \times \frac{3}{2} =- \frac{9}{4}$ . The lines are neither parallel nor perpendicular.

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Example Question #1 : How To Find Out If Lines Are Perpendicular

The slopes of two lines on the coordinate plane are -0.25 and 4.

True or false: the lines are perpendicular.

Possible Answers:

True

False

Correct answer:

True

Explanation:

Two lines on the coordinate plane are perpendicular if and only if the product of their slopes is . The product of the slopes of the lines in question is

$- 0.25 \cdot 4 = -1$ ,

so the lines are indeed perpendicular.

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Example Question #11 : How To Find Out If Lines Are Perpendicular

Two lines intersect at the point (3, 8) . One line passes through the point (7, 3) ; the other passes through (-2, 12) .

True or false: The lines are perpendicular.

Possible Answers:

True

False

Correct answer:

False

Explanation:

Two lines are perpendicular if and only if the product of their slopes is . The slope of each line can be found from the coordinates of two points using the slope formula

$m = \frac{y_{2}- y_{1}}{x_{2}-x_{1}}$

To find the slope of the first line, set $x_{1} = 3,y_{1}= 8 , x_{2} = 7, y_{2}= 3$ :

$m = \frac{3-8}{7-3} = \frac{-5}{4} = -\frac{5}{4}$

To find the slope of the second line, set $x_{1} = 3,y_{1}= 8 , x_{2} = -2, y_{2}=1 2$ :

$m = \frac{12-8}{-2-3} = \frac{4}{-5} = -\frac{4}{5}$

The product of the slopes is

$-\frac{5}{4} \left ( -\frac{4}{5} \right )= 1$

As the product is not , the lines are not perpendicular.

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Example Question #171 : Coordinate Geometry

The slopes of two lines are 6 and - 0.16666 . True or false: the lines are perpendicular.

Possible Answers:

False

True

Correct answer:

False

Explanation:

Two lines on the coordinate plane are perpendicular if and only if the product of their slopes is . The product of the slopes of the lines in question is

$- 0.16666 \cdot 6 = -0.99996$

The product is not equal to , so the lines are not perpendicular.

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Intermediate Geometry : Perpendicular Lines

Study concepts, example questions & explanations for Intermediate Geometry

All Intermediate Geometry Resources

Example Questions

Example Question #8 : How To Find Out If Lines Are Perpendicular

Example Question #9 : How To Find Out If Lines Are Perpendicular

Example Question #1 : How To Find Out If Lines Are Perpendicular

Example Question #11 : How To Find Out If Lines Are Perpendicular

Example Question #171 : Coordinate Geometry

All Intermediate Geometry Resources

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