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.

2.
Are the two lines $y$ = 2$x$ + 3 and $y$ = 2$x$ + 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.

3.
Are the two lines $y$ = 3$x$ + 2 and $y$ = -4$x$ + $\frac{2}{3}$ 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.

4.
Is the line passing through the points (2, 3) and (-4, -5) perpendicular to the line $y$ = $\frac{3x}{4}$ + 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.

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.

6.
Write an equation of the line passing through the point (-3, -4) and perpendicular to the line $y$ = 5$x$ + 3.
 a. $y$ = - $\frac{1}{5}$($x$ + 3) b. $y$ + 4 = - $\frac{1}{5}$($x$ + 3) c. $y$ + 4 = $\frac{1}{5}$($x$ + 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).

7.
Which among the following are not the slopes of two perpendicular lines?
 a. 3 & $\frac{-1}{3}$ b. 1 & -1 c. $\frac{3}{4}$ & $\frac{-4}{3}$ d. 8 & $\frac{1}{8}$

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.

8.
Is the line passing through the points (-4, 5) and (-5, 9) perpendicular to the line $y$ = $\frac{x}{4}$ + 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.

9.
Are the two lines $y$ = 5$x$ + 4 and $y$ = -5$x$ + 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.

10.
Are the two lines $y$ = -5$x$ - 5 and $y$ = -5$x$ + 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.