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Perpendicular Lines Worksheets

Perpendicular Lines Worksheets
  • Page 1
 1.  
Is the line passing through the points (-3, 4) and (-5, 6) perpendicular to the line y = x + 5?
a.
Yes
b.
No


Solution:

Slope of a line passing through the points (x1, y1) and (x2, y2) is (y2 - y1) / (x2 - x1).

Slope of the line passing through the points (-3, 4) and (-5, 6) = 6-4 / -5-(-3)

= -1
[Simplify.]

Slope of the line y = x + 5 is 1.

Product of the slopes of the two lines = -1 x 1 = -1
[Multiply.]

Two lines are perpendicular, only if the product of their slopes is -1.

So, the two lines are perpendicular.


Correct answer : (1)
 2.  
Are the two lines y = 2x + 3 and y = 2x + 4 perpendicular?
a.
Yes
b.
No


Solution:

Two lines are perpendicular, only if the product of their slopes is equal to -1.

The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.

The two given lines are in slope-intercept form.

Slope of the line y = 2x + 3 is 2.
[Compare with the equation in step 1.]

Slope of the line y = 2x + 4 is 2.
[Compare with the equation in step 1.]

Product of the slopes of the two lines = 2 x 2 = 4
[Multiply.]

So, the two equations are not perpendicular.


Correct answer : (2)
 3.  
Are the two lines y = 3x + 2 and y = -4x + 2 3 perpendicular?
a.
No
b.
Yes


Solution:

The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.

The two given lines are in slope-intercept form.

Slope of the line y = 3x + 2 is 3.
[Compare with the equation in step 1.]

Slope of the line y = -4x + 2 / 3 is -4.
[Compare with the equation in step 1.]

Product of the slopes of two lines = 3 x (-4) = -12
[Multiply.]

Two lines are perpendicular, if the product of the slopes of the two lines is equal to -1.

The product of the slopes of the two lines is -12. So, they are not perpendicular.


Correct answer : (1)
 4.  
Is the line passing through the points (2, 3) and (-4, -5) perpendicular to the line y = 3x 4 + 4?
a.
No
b.
Yes


Solution:

Slope of the line passing through the points (x1, y1) and (x2, y2) is (y2 - y1) / (x2 - x1).

So, the slope of the line passing through (2, 3) and (-4, -5) = (-5-3) / (-4-2) = 4 / 3
[Substitute x1 = 2, y1 = 3, x2 = -4 and y2 = -5 and simplify.]

Slope of the line y = 3x / 4 + 4 is 3 / 4.
[Slope = coefficient of x.]

Product of the slopes of the two lines = 4 / 3 x 3 / 4 = 1
[Multiply.]

Two lines are perpendicular, if the product of their slopes is -1.

The product of the slopes of the two lines is 1. So, the two lines are not perpendicular.


Correct answer : (1)
 5.  
___ and ___ lines are always perpendicular to each other.
a.
Intersecting, non-intersecting
b.
Horizontal, vertical
c.
Parallel, skew
d.
Parallel, transversal


Solution:

The angle formed between horizontal and vertical lines is always 90o.

So, horizontal and vertical lines are perpendicular to each other.


Correct answer : (2)
 6.  
Write an equation of the line passing through the point (-3, -4) and perpendicular to the line y = 5x + 3.
a.
y = - 1 5(x + 3)
b.
y + 4 = - 1 5(x + 3)
c.
y + 4 = 1 5(x + 3)
d.
None of the above


Solution:

The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.

Slope of the line y = 5x + 3 is 5.
[Compare with the equation in step 1.]

Slope of the line perpendicular to y = 5x + 3 is - 1 / 5.
[Product of the slopes of perpendicular lines is -1.]

The equation of the line passing through the point (x1, y1) with slope m in point-slope form is y - y1 = m(x - x1).

y - (-4) = - 15[x - (-3)]
[Substitute (x1, y1) = (-3, -4) and m = - 1 / 5 in the equation in step 4.]

y + 4 = - 15(x + 3)
[Simplify the equation.]

The equation of the line passing through the point (-3, -4) is y + 4 = - 1 / 5(x + 3).


Correct answer : (2)
 7.  
Which among the following are not the slopes of two perpendicular lines?
a.
3 & -1 3
b.
1 & -1
c.
3 4 & -4 3
d.
8 & 1 8


Solution:

Product of slopes of two perpendicular lines is -1.

Among the choices, 8 x 1 / 8 = 1 ≠ -1
[Multiply and simplify.]

So, 8 and 1 / 8 are not the slopes of two perpendicular lines.


Correct answer : (4)
 8.  
Is the line passing through the points (-4, 5) and (-5, 9) perpendicular to the line y = x 4 + 5?
a.
Yes
b.
No


Solution:

Slope of the line passing through the points (x1, y1) and (x2, y2) is (y2 - y1) / (x2 - x1).


The slope of the line passing through the points (-4, 5) and (-5, 9) is (9-5) / [(-5)-(-4)]
[Substitute x1 = -4, y1 = 5, x2 = -5 and y2 = 9]

= 4-1 = -4
[Simplify the fraction.]

Slope of the line y = x / 4 + 5 is 1 / 4.
[Slope = coefficient of x.]

Product of the slopes of the two lines = -4 x 1 / 4 = -1
[Multiply.]

Two lines are perpendicular, if the product of their slopes is -1.

So, the two lines are perpendicular.


Correct answer : (1)
 9.  
Are the two lines y = 5x + 4 and y = -5x + 7 perpendicular?
a.
Yes
b.
No


Solution:

The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.

The two given lines are in slope-intercept form.

Slope of the line y = 5x + 4 is 5
[Compare with the equation in step 1.]

Slope of the line y = -5x + 7 is -5
[Compare with the equation in step 1.]

Product of the slopes of the two lines = 5 x -5 = -25
[Multiply.]

Two lines are perpendicular, only if the product of their slopes is -1.

The product of the slopes of the two lines is -25. So, they are not perpendicular.


Correct answer : (2)
 10.  
Are the two lines y = -5x - 5 and y = -5x + 5 perpendicular?
a.
Yes
b.
No


Solution:

The slope-intercept form of the equation of a line with slope m and y-intercept b is y = mx + b.

The two given lines are in slope-intercept form.

Slope of the line y = -5x - 5 is -5
[Compare with the equation in step 1.]

Slope of the line y = -5x + 5 is -5
[Compare with the equation in step 1.]

Product of the slopes of the two lines = -5 x -5 = 25
[Multiply.]

The product of the slopes of the two lines is not equal to -1.

So, the two equations are not perpendicular.


Correct answer : (2)

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Perpendicular Lines Worksheet
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