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Worksheet: Distance Between a Point and a Straight Line in the Coordinate Plane

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

Q1:

Find the length of the perpendicular drawn from the origin to the straight line βˆ’ 3 π‘₯ + 4 𝑦 βˆ’ 2 1 = 0 rounded to the nearest hundredth.

Q2:

Find the length of the perpendicular line drawn from the point to the straight line that passes through the point and whose gradient is .

  • A length units
  • B length units
  • C length units
  • D length units

Q3:

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

Q4:

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

Q5:

Find the length of the perpendicular drawn from the point 𝐴 ( 8 , βˆ’ 2 ) to the straight line passing through the points 𝐡 ( βˆ’ 7 , βˆ’ 6 ) and 𝐢 ( 9 , 6 ) .

  • A 5 8 √ 1 7 1 7 units length
  • B 5 2 9 units length
  • C 3 2 5 units length
  • D 2 9 5 units length

Q6:

Find the length of the perpendicular drawn from the point 𝐴 ( βˆ’ 8 , 1 0 ) to the straight line passing through the points 𝐡 ( βˆ’ 3 , βˆ’ 2 ) and 𝐢 ( βˆ’ 8 , 6 ) .

  • A 1 0 √ 4 1 4 1 units length
  • B √ 8 9 2 0 units length
  • C 1 4 √ 8 9 8 9 units length
  • D 2 0 √ 8 9 8 9 units length

Q7:

If the length of the perpendicular drawn from the point ( βˆ’ 5 , 𝑦 ) to the straight line βˆ’ 1 5 π‘₯ + 8 𝑦 βˆ’ 5 = 0 is 10 length units, find all the possible values of 𝑦 .

  • A 𝑦 = βˆ’ 3 0 or 𝑦 = 3 0
  • B 𝑦 = βˆ’ 2 5 2 or 𝑦 = 2 5 2
  • C 𝑦 = βˆ’ 4 3 3 or 𝑦 = 2 5 3
  • D 𝑦 = βˆ’ 3 0 or 𝑦 = 2 5 2

Q8:

If the length of the perpendicular drawn from the point ( βˆ’ 8 , 𝑦 ) to the straight line 3 π‘₯ + 4 𝑦 + 4 = 0 is 8 length units, find all the possible values of 𝑦 .

  • A 𝑦 = βˆ’ 5 or 𝑦 = 5
  • B 𝑦 = βˆ’ 1 5 or 𝑦 = 1 5
  • C 𝑦 = βˆ’ 4 or 𝑦 = 6 8 3
  • D 𝑦 = βˆ’ 5 or 𝑦 = 1 5

Q9:

If the length of the perpendicular drawn from the point ( 8 , 𝑦 ) to the straight line βˆ’ 3 π‘₯ βˆ’ 4 𝑦 + 5 = 0 is 6 length units, find all the possible values of 𝑦 .

  • A 𝑦 = βˆ’ 1 1 4 or 𝑦 = 1 1 4
  • B 𝑦 = βˆ’ 4 9 4 or 𝑦 = 4 9 4
  • C 𝑦 = βˆ’ 1 9 or 𝑦 = 1
  • D 𝑦 = βˆ’ 4 9 4 or 𝑦 = 1 1 4

Q10:

If the length of the perpendicular drawn from the point ( 7 , 𝑦 ) to the straight line 1 2 π‘₯ βˆ’ 5 𝑦 + 4 = 0 is 9 length units, find all the possible values of 𝑦 .

  • A 𝑦 = βˆ’ 4 1 or 𝑦 = 4 1
  • B 𝑦 = βˆ’ 2 9 5 or 𝑦 = 2 9 5
  • C 𝑦 = βˆ’ 4 3 6 or 𝑦 = 3 7 3
  • D 𝑦 = βˆ’ 2 9 5 or 𝑦 = 4 1

Q11:

Find the length of the perpendicular from the point ( βˆ’ 2 2 , βˆ’ 5 ) to the π‘₯ -axis.

Q12:

Find the length of the perpendicular from the point ( 2 9 , 1 1 ) to the π‘₯ -axis.

Q13:

Find all values of π‘Ž for which the distance between the line π‘Ž π‘₯ + 𝑦 βˆ’ 7 = 0 and point ( βˆ’ 4 , 3 ) is 2 0 √ 8 2 4 1 .

  • A βˆ’ 3 6 or 3
  • B βˆ’ 9 or βˆ’ 1 9
  • C18 or 2 9
  • D9 or 1 9

Q14:

Find all values of π‘Ž for which the distance between the line π‘Ž π‘₯ βˆ’ 4 𝑦 + 4 = 0 and point ( 3 , βˆ’ 1 ) is 5 √ 1 7 1 7 .

  • A βˆ’ 3 or 4
  • B 4 3 8 or 1
  • C βˆ’ 4 3 4 or βˆ’ 2
  • D βˆ’ 4 3 8 or βˆ’ 1

Q15:

Find all values of π‘Ž for which the distance between the line π‘Ž π‘₯ βˆ’ 6 𝑦 + 4 = 0 and point ( 2 , 0 ) is 6 √ 6 1 6 1 .

  • A βˆ’ 1 0 or 0
  • B5 or βˆ’ 4 1 3
  • C βˆ’ 1 0 or 8 1 3
  • D βˆ’ 5 or 4 1 3

Q16:

Find all values of π‘Ž for which the distance between the line π‘Ž π‘₯ βˆ’ 𝑦 + 4 = 0 and point ( βˆ’ 4 , βˆ’ 3 ) is √ 2 6 2 .

  • A βˆ’ 2 0 or 3
  • B βˆ’ 1 7 1 9 or βˆ’ 5
  • C 3 4 1 9 or 10
  • D 1 7 1 9 or 5

Q17:

Determine the length of the perpendicular from a point 𝐴 ( π‘₯ , 𝑦 ) 1 1 to the line 𝑦 = 0 .

  • A0
  • B | π‘₯ | 1
  • C  | π‘₯ | + | 𝑦 | 1 2 1 2
  • D | 𝑦 | 1
  • E | 𝑦 | | π‘₯ | 1 1

Q18:

What is the distance between the point and the line of gradient 1 through ?

  • A length units
  • B length units
  • C length units
  • D length units

Q19:

Find the length of the perpendicular drawn from the point 𝐴 ( 1 , 9 ) to the straight line βˆ’ 5 π‘₯ + 1 2 𝑦 + 1 3 = 0 .

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

Q20:

Find the length of the perpendicular drawn from the point 𝐴 ( βˆ’ 3 , 5 ) to the straight line 4 π‘₯ βˆ’ 2 𝑦 + 7 = 0 .

  • A 3 4 length units
  • B 9 √ 5 1 0 length units
  • C 5 √ 6 2 length units
  • D 3 √ 5 2 length units

Q21:

Find the length of the perpendicular drawn from the point 𝐴 ( βˆ’ 6 , βˆ’ 8 ) to the straight line 8 π‘₯ + 𝑦 + 1 2 = 0 .

  • A 4 4 6 5 length units
  • B 4 √ 6 5 5 length units
  • C 4 4 3 length units
  • D 4 4 √ 6 5 6 5 length units

Q22:

Find the length of the perpendicular drawn from the point 𝐴 ( 3 , 7 ) to the straight line βˆ’ 4 π‘₯ + 9 𝑦 + 6 = 0 .

  • A 5 7 9 7 length units
  • B 8 1 √ 9 7 9 7 length units
  • C 5 7 √ 1 3 1 3 length units
  • D 5 7 √ 9 7 9 7 length units

Q23:

Find the length of the perpendicular drawn from the point 𝐴 ( 2 , 6 ) to the straight line π‘₯ + 2 𝑦 + 1 0 = 0 .

  • A 2 4 5 length units
  • B 4 √ 5 length units
  • C 8 √ 3 length units
  • D 2 4 √ 5 5 length units

Q24:

Find the length of the perpendicular from the point ( βˆ’ 1 9 , βˆ’ 1 3 ) to the 𝑦 -axis.

Q25:

Find the length of the perpendicular from the point ( 1 8 , 1 1 ) to the 𝑦 -axis.

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