Lesson Worksheet: Perpendicular Distance from a Point to a Line on the Coordinate Plane Mathematics

In this worksheet, we will practice finding the perpendicular distance between a point and a straight line or between two parallel lines on the coordinate plane using the formula.

Q1:

Determine the length of the perpendicular from a point 𝐴(𝑥,𝑦) to the line 𝑦=0.

  • A|𝑦|
  • B|𝑦||𝑥|
  • C|𝑥|
  • D|𝑥|+|𝑦|
  • E0

Q2:

Find the length of the perpendicular drawn from the origin to the straight line −3𝑥+4𝑦−21=0 rounded to the nearest hundredth.

  • A4.20 length units
  • B14.85 length units
  • C0.24 length units
  • D21.00 length units

Q3:

Find the length of the perpendicular from the point (−22,−5) to the 𝑥-axis.

Q4:

Find the length of the perpendicular from the point (−19,−13) to the 𝑦-axis.

Q5:

Find the length of the perpendicular drawn from the point 𝐴(1,9) to the straight line −5𝑥+12𝑦+13=0.

  • A116√1717 length units
  • B11613 length units
  • C116169 length units
  • D12613 length units

Q6:

Find the length of the perpendicular drawn from the point 𝐴(−1,−7) to the straight line passing through the points 𝐵(6,−4) and 𝐶(9,−5).

  • A11√105 units length
  • B√1016 units length
  • C8√105 units length
  • D8√25 units length

Q7:

If the length of the perpendicular drawn from the point (−5,𝑦) to the straight line −15𝑥+8𝑦−5=0 is 10 length units, find all the possible values of 𝑦.

  • A𝑦=−433 or 𝑦=253
  • B𝑦=−30 or 𝑦=30
  • C𝑦=−252 or 𝑦=252
  • D𝑦=−30 or 𝑦=252

Q8:

Find all values of 𝑎 for which the distance between the line 𝑎𝑥+𝑦−7=0 and point (−4,3) is 20√8241.

  • A18 or 29
  • B−36 or 3
  • C9 or 19
  • D−9 or −19

Q9:

Find the length of the perpendicular line drawn from the point 𝐴(−8,5) to the straight line that passes through the point 𝐵(2,−4) and whose slope is =−8.

  • A4965 length units
  • B71√6565 length units
  • C62√6565 length units
  • D718 length units

Q10:

What is the distance between the point (−9,−10) and the line of slope 1 through (3,−7)?

  • A29√22 length units
  • B23√22 length units
  • C9√22 length units
  • D5√22 length units

Q11:

Find the perpendicular distance between the point 𝐴(2,20) and the 𝑥-axis.

Q12:

If the length of the perpendicular drawn from the point 𝐴(7,−1) to the straight line −5𝑥−2𝑦+𝑐=0 equals 24√2929, find all possible values of 𝑐.

  • A83 or −9
  • B−57 or −9
  • C−9 or −61
  • D83 or −61
  • E57 or 9

Q13:

Suppose 𝐴𝐵 and 𝐴𝐶 are equal chords in a circle 𝑀, where the coordinates of points 𝑀, 𝐴, and 𝐵 are (−9,0), (−11,−2), and (−7,−2) respectively. Find the distance between chord 𝐴𝐶 and 𝑀.

  • A2
  • B2√2
  • C4√2
  • D4

Q14:

What is the distance between lines (−16,−16)+𝑘(2,4) and (19,−17)+𝑘(7,14)?

  • A71√33
  • B71√55
  • C37√55
  • D715

Q15:

Find the shortest distance between the line 𝑦=1 and point 𝐴(1,7).

Q16:

Find the shortest distance between the line 𝑦=12𝑥−2 and the point 𝐴(9,−10).

  • A√5
  • B5√5
  • C√3
  • D√277
  • E√11

Q17:

Find the shortest distance between the point (−6,10) and the line which passes through the points (1,9) and (4,6).

  • A4√2
  • B3√2
  • C√14
  • D7√2
  • E√6

Q18:

Determine the shortest distance between the line 𝑥=3 and the point 𝐴(−8,−6).

Q19:

Find the length of the perpendicular drawn from the point 𝐴(−9,5) to the straight line passing through the points 𝐵(4,3) and 𝐶(−2,−7).

  • A30√3417 units length
  • B√3471 units length
  • C71√3434 units length
  • D71√10653 units length

Q20:

The length of the perpendicular from the point 𝐴(−1,−1) to a straight line is 2 length units, where the vector (−9,−12) is the direction vector of the straight line. Determine the equations of the two straight lines that satisfy these conditions.

  • A4𝑥−3𝑦+11=0, 4𝑥−3𝑦−9=0
  • B4𝑥−3𝑦+9=0, 4𝑥−3𝑦−9=0
  • C3𝑥−4𝑦−9=0, 3𝑥−4𝑦+11=0
  • D3𝑥+4𝑦+17=0, 3𝑥+4𝑦−3=0

Q21:

Calculate the area of the circle of center (−8,14) that is tangent to the line −5𝑥−12𝑦+9=0. Give your answer to the nearest hundredth.

Q22:

Determine the shortest distance between the two parallel lines whose equations are 𝑦=4 and 𝑦=−4.

Q23:

Find the shortest distance between the two parallel lines with equations 𝑦=2𝑥−7 and 𝑦=2𝑥+3.

  • A√2
  • B2√5
  • C5√2
  • D4√2
  • E2√2

Q24:

The distance between two parallel lines is √37. If a third line is perpendicuar to both these lines, and intersects them at the points 𝐴(𝑎,2) and 𝐵(−10,3), find all possible values of 𝑎.

  • A𝑎=46 or 𝑎=−26
  • B𝑎=−4 or 𝑎=−16
  • C𝑎=26 or 𝑎=−46
  • D𝑎=−4 or 𝑎=−16
  • E𝑎=16 or 𝑎=4

Q25:

Find the shortest distance between the following two parallel lines: 𝑦=3𝑥+5,𝑦=3𝑥−5. If necessary, round your answer to 2 decimal places.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.