Practice Parallel and Perpendicular in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Parallel lines never intersect and have matching direction; perpendicular lines intersect at right angles.

Parallel tracks run side by side; perpendicular streets form a plus sign.

Showing a random 20 of 50 problems.

Example 1

easy
Two streets cross to form a plus sign. What is their geometric relationship?

Example 2

medium
Determine whether the quadrilateral with vertices A(0,0)A(0,0), B(4,0)B(4,0), C(5,3)C(5,3), D(1,3)D(1,3) is a parallelogram.

Example 3

medium
For what value of k are y=kxy = kx and y=3x+1y = 3x + 1 perpendicular?

Example 4

easy
What is true about the slopes of two parallel lines?

Example 5

easy
Are y=12x+4y=\tfrac{1}{2}x+4 and y=12x−7y=\tfrac{1}{2}x-7 parallel, perpendicular, or neither?

Example 6

hard
Find kk so that lines y=kx+1y=kx+1 and y=(k−2)x−3y=(k-2)x-3 are perpendicular.

Example 7

medium
Find the equation of the line through (2,−7)(2,-7) parallel to y=−3x+1y=-3x+1.

Example 8

medium
Find the equation of the line through (0,2)(0,2) perpendicular to y=12x+5y = \frac{1}{2}x + 5.

Example 9

medium
Lines 2x+3y=62x + 3y = 6 and 3x−2y=43x - 2y = 4 are related how?

Example 10

challenge
Explain why a line perpendicular to one of two parallel lines is perpendicular to both.

Example 11

medium
Line ℓ\ell passes through (0,3)(0,3) and (6,0)(6,0). Find the equation of the line perpendicular to ℓ\ell passing through the origin.

Example 12

easy
Are the lines y=2x+1y = 2x + 1 and y=2x−5y = 2x - 5 parallel, perpendicular, or neither?

Example 13

challenge
Show that the points A(1,2)A(1,2), B(4,3)B(4,3), C(5,6)C(5,6), D(2,5)D(2,5) form a rhombus (all sides equal) but not a square (no right angle).

Example 14

easy
What is the slope of a line perpendicular to a horizontal line?

Example 15

hard
The line ℓ\ell passes through (2,1)(2,1) and (6,4)(6,4). Find the equation of the line through (0,0)(0,0) perpendicular to ℓ\ell.

Example 16

medium
What is the slope of any line parallel to the xx-axis?

Example 17

medium
Find the equation of the line through (0,−2)(0,-2) parallel to 2x+5y=102x+5y=10.

Example 18

easy
Find the equation of the line through (3,−1)(3, -1) that is parallel to y=2x+5y = 2x + 5.

Example 19

hard
Find the equation of the perpendicular bisector of the segment from (2,1)(2,1) to (8,5)(8,5).

Example 20

medium
Find the equation of the line through (−3,2)(-3,2) perpendicular to y=14x−7y=\tfrac{1}{4}x-7.
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