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+4𝑦21=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.

  • A1161717 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).

  • A11105 units length
  • B1016 units length
  • C8105 units length
  • D825 units length

Q7:

If the length of the perpendicular drawn from the point (5,𝑦) to the straight line 15𝑥+8𝑦5=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+𝑦7=0 and point (4,3) is 208241.

  • A18 or 29
  • B36 or 3
  • C9 or 19
  • D9 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
  • B716565 length units
  • C626565 length units
  • D718 length units

Q10:

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

  • A2922 length units
  • B2322 length units
  • C922 length units
  • D522 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 5𝑥2𝑦+𝑐=0 equals 242929, find all possible values of 𝑐.

  • A83 or 9
  • B57 or 9
  • C9 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
  • B22
  • C42
  • D4

Q14:

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

  • A7133
  • B7155
  • C3755
  • 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=12𝑥2 and the point 𝐴(9,10).

  • A5
  • B55
  • C3
  • D277
  • E11

Q17:

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

  • A42
  • B32
  • C14
  • D72
  • E6

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).

  • A303417 units length
  • B3471 units length
  • C713434 units length
  • D7110653 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.

  • A4𝑥3𝑦+11=0, 4𝑥3𝑦9=0
  • B4𝑥3𝑦+9=0, 4𝑥3𝑦9=0
  • C3𝑥4𝑦9=0, 3𝑥4𝑦+11=0
  • D3𝑥+4𝑦+17=0, 3𝑥+4𝑦3=0

Q21:

Calculate the area of the circle of center (8,14) that is tangent to the line 5𝑥12𝑦+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=2𝑥7 and 𝑦=2𝑥+3.

  • A2
  • B25
  • C52
  • D42
  • E22

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=3𝑥5. If necessary, round your answer to 2 decimal places.

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