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.

  • A 1 1 6 √ 1 7 1 7 length units
  • B 1 1 6 1 3 length units
  • C 1 1 6 1 6 9 length units
  • D 1 2 6 1 3 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).

  • A 1 1 √ 1 0 5 units length
  • B √ 1 0 1 6 units length
  • C 8 √ 1 0 5 units length
  • D 8 √ 2 5 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.

  • A 𝑦 = βˆ’ 4 3 3 or 𝑦=253
  • B 𝑦 = βˆ’ 3 0 or 𝑦=30
  • C 𝑦 = βˆ’ 2 5 2 or 𝑦=252
  • D 𝑦 = βˆ’ 3 0 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
  • B βˆ’ 3 6 or 3
  • C9 or 19
  • D βˆ’ 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.

  • A 4 9 6 5 length units
  • B 7 1 √ 6 5 6 5 length units
  • C 6 2 √ 6 5 6 5 length units
  • D 7 1 8 length units

Q10:

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

  • A 2 9 √ 2 2 length units
  • B 2 3 √ 2 2 length units
  • C 9 √ 2 2 length units
  • D 5 √ 2 2 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
  • B βˆ’ 5 7 or βˆ’9
  • C βˆ’ 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
  • B 2 √ 2
  • C 4 √ 2
  • D4

Q14:

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

  • A 7 1 √ 3 3
  • B 7 1 √ 5 5
  • C 3 7 √ 5 5
  • D 7 1 5

Q15:

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

  • A 4 √ 4 1 length units
  • B20 length units
  • C 4 √ 3 4 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
  • B 5 √ 5
  • C √ 3
  • D √ 2 7 7
  • E √ 1 1

Q18:

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

  • A 4 √ 2
  • B 3 √ 2
  • C √ 1 4
  • D 7 √ 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).

  • A 3 0 √ 3 4 1 7 units length
  • B √ 3 4 7 1 units length
  • C 7 1 √ 3 4 3 4 units length
  • D 7 1 √ 1 0 6 5 3 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.

  • A 4 π‘₯ βˆ’ 3 𝑦 + 1 1 = 0 , 4 π‘₯ βˆ’ 3 𝑦 βˆ’ 9 = 0
  • B 4 π‘₯ βˆ’ 3 𝑦 + 9 = 0 , 4 π‘₯ βˆ’ 3 𝑦 βˆ’ 9 = 0
  • C 3 π‘₯ βˆ’ 4 𝑦 βˆ’ 9 = 0 , 3 π‘₯ βˆ’ 4 𝑦 + 1 1 = 0
  • D 3 π‘₯ + 4 𝑦 + 1 7 = 0 , 3 π‘₯ + 4 𝑦 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
  • B 2 √ 5
  • C 5 √ 2
  • D 4 √ 2
  • E 2 √ 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.

  • A π‘Ž = 4 6 or π‘Ž=βˆ’26
  • B π‘Ž = βˆ’ 4 or π‘Ž=βˆ’16
  • C π‘Ž = 2 6 or π‘Ž=βˆ’46
  • D π‘Ž = βˆ’ 4 or π‘Ž=βˆ’16
  • E π‘Ž = 1 6 or π‘Ž=4

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