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

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In this worksheet, we will practice finding the perpendicular distance between a point and a straight line in the coordinate plane using the formula.

Q1:

Determine the length of the perpendicular from a point 𝐴(π‘₯,𝑦) to the line 𝑦=0.

  • A|𝑦|
  • B|𝑦||π‘₯|
  • C|π‘₯|
  • D|π‘₯|+|𝑦|
  • E0

Q2:

Find the length of the perpendicular drawn from the origin to the straight line βˆ’3π‘₯+4π‘¦βˆ’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 (βˆ’22,βˆ’5) to the π‘₯-axis.

Q4:

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

Q5:

Find the length of the perpendicular drawn from the point 𝐴(1,9) to the straight line βˆ’5π‘₯+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(βˆ’1,βˆ’7) to the straight line passing through the points 𝐡(6,βˆ’4) and 𝐢(9,βˆ’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 (βˆ’5,𝑦) to the straight line βˆ’15π‘₯+8π‘¦βˆ’5=0 is 10 length units, find all the possible values of 𝑦.

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

Q8:

Find all values of π‘Ž for which the distance between the line π‘Žπ‘₯+π‘¦βˆ’7=0 and point (βˆ’4,3) is 20√8241.

  • A18 or 29
  • Bβˆ’36 or 3
  • C9 or 19
  • Dβˆ’9 or βˆ’19

Q9:

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

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

Q10:

What is the distance between the point (βˆ’9,βˆ’10) and the line of slope 1 through (3,βˆ’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 π‘₯-axis.

Q12:

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

  • A83 or βˆ’9
  • Bβˆ’57 or βˆ’9
  • Cβˆ’9 or βˆ’61
  • D83 or βˆ’61
  • E57 or 9

Q13:

Suppose 𝐴𝐡 and 𝐴𝐢 are equal chords in a circle 𝑀, where the coordinates of points 𝑀, 𝐴, and 𝐡 are (βˆ’9,0), (βˆ’11,βˆ’2), and (βˆ’7,βˆ’2), respectively. Find the distance between chord 𝐴𝐢 and 𝑀.

  • A2
  • B2√2
  • C4√2
  • D4

Q14:

What is the distance between lines (βˆ’16,βˆ’16)+π‘˜(2,4) and (19,βˆ’17)+π‘˜(7,14)?

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

Q15:

What is the distance between the point (16,12,20) and the 𝑦-axis?

  • A4√41 length units
  • B20 length units
  • C4√34 length units
  • D6 length units

Q16:

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

Q17:

Find the shortest distance between the line 𝑦=12π‘₯βˆ’2 and the point 𝐴(9,βˆ’10).

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

Q18:

Find the shortest distance between the point (βˆ’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

Q19:

Determine the shortest distance between the line π‘₯=3 and the point 𝐴(βˆ’8,βˆ’6).

Q20:

Find the length of the perpendicular drawn from the point 𝐴(βˆ’9,5) to the straight line passing through the points 𝐡(4,3) and 𝐢(βˆ’2,βˆ’7).

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

Q21:

If the length of the perpendicular from the point 𝐴(βˆ’1,βˆ’1) to a straight line is 2 length units, where the vector (βˆ’9,βˆ’12) is the direction vector of the straight line. Determine the equation of the straight line.

  • A4π‘₯βˆ’3𝑦+11=0, 4π‘₯βˆ’3π‘¦βˆ’9=0
  • B4π‘₯βˆ’3𝑦+9=0, 4π‘₯βˆ’3π‘¦βˆ’9=0
  • C3π‘₯βˆ’4π‘¦βˆ’9=0, 3π‘₯βˆ’4𝑦+11=0
  • D3π‘₯+4𝑦+17=0, 3π‘₯+4π‘¦βˆ’3=0

Q22:

Calculate the area of the circle of center (βˆ’8,14) that is tangent to the line βˆ’5π‘₯βˆ’12𝑦+9=0. Give your answer to the nearest hundredth.

Q23:

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

Q24:

Find the shortest distance between the two parallel lines with equations 𝑦=2π‘₯βˆ’7 and 𝑦=2π‘₯+3.

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

Q25:

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(π‘Ž,2) and 𝐡(βˆ’10,3), find all possible values of π‘Ž.

  • Aπ‘Ž=46 or π‘Ž=βˆ’26
  • Bπ‘Ž=βˆ’4 or π‘Ž=βˆ’16
  • Cπ‘Ž=26 or π‘Ž=βˆ’46
  • Dπ‘Ž=βˆ’4 or π‘Ž=βˆ’16
  • Eπ‘Ž=16 or π‘Ž=4

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