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

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

Q1:

Determine the length of the perpendicular from a point 𝐴 ( 𝑥 , 𝑦 ) 1 1 to the line 𝑦 = 0 .

  • A0
  • B | 𝑥 | 1
  • C | 𝑥 | + | 𝑦 | 1 2 1 2
  • D | 𝑦 | 1
  • E | 𝑦 | | 𝑥 | 1 1

Q2:

Find the length of the perpendicular drawn from the origin to the straight line − 3 𝑥 + 4 𝑦 − 2 1 = 0 rounded to the nearest hundredth.

Q3:

Find the length of the perpendicular from the point ( − 2 2 , − 5 ) to the 𝑥 -axis.

Q4:

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

Q5:

Find the length of the perpendicular drawn from the point 𝐴 ( 1 , 9 ) to the straight line − 5 𝑥 + 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

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 ) .

  • 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

Q7:

If the length of the perpendicular drawn from the point ( − 5 , 𝑦 ) to the straight line − 1 5 𝑥 + 8 𝑦 − 5 = 0 is 10 length units, find all the possible values of 𝑦 .

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

Q8:

Find all values of 𝑎 for which the distance between the line 𝑎 𝑥 + 𝑦 − 7 = 0 and point ( − 4 , 3 ) is 2 0 √ 8 2 4 1 .

  • A − 3 6 or 3
  • B − 9 or − 1 9
  • C18 or 2 9
  • D9 or 1 9

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 .

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

Q10:

What is the distance between the point ( − 9 , − 1 0 ) and the line of slope 1 through ( 3 , − 7 ) ?

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

Q11:

Find the perpendicular distance between the point 𝐴 ( 2 , 2 0 ) and the 𝑥 -axis.

Q12:

If the length of the perpendicular drawn from the point 𝐴 ( 7 , − 1 ) to the straight line − 5 𝑥 − 2 𝑦 + 𝑐 = 0 equals 2 4 √ 2 9 2 9 , find all possible values of 𝑐 .

  • A − 9 or − 6 1
  • B − 5 7 or − 9
  • C83 or − 9
  • D57 or 9
  • E83 or − 6 1

Q13:

Suppose 𝐴 𝐵 and 𝐴 𝐶 are equal chords in a circle 𝑀 , where the coordinates of points 𝑀 , 𝐴 , and 𝐵 are ( − 9 , 0 ) , ( − 1 1 , − 2 ) , and ( − 7 , − 2 ) , respectively. Find the distance between chord 𝐴 𝐶 and 𝑀 .

  • A4
  • B 2 √ 2
  • C 4 √ 2
  • D2

Q14:

Let 𝐿 be the line through point ( 7 , 5 , 5 ) in the direction of vector ( 2 , 4 , − 9 ) . Find the distance between 𝐿 and the point ( 2 , 6 , 6 ) , to the nearest hundredth.

Q15:

Let 𝐿 be the line through the point ( − 6 , 8 , 9 ) that makes equal angles with the three coordinate axes. What is the distance between the point ( − 4 , 5 , 3 ) and 𝐿 , to the nearest hundredth.

Q16:

Determine, to the nearest hundredth, the length of the perpendicular drawn from the point ( − 5 , − 7 , − 1 0 ) to the straight line 𝑥 + 8 2 = 𝑦 − 9 8 = 𝑧 + 7 − 8 .

Q17:

What is the distance between lines ( − 1 6 , − 1 6 ) + 𝑘 ( 2 , 4 ) and ( 1 9 , − 1 7 ) + 𝑘 ( 7 , 1 4 ) ?

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

Q18:

What is the distance between the point ( 1 6 , 1 2 , 2 0 ) and the 𝑦 -axis?

  • A 4 √ 3 4 length units
  • B20 length units
  • C6 length units
  • D 4 √ 4 1 length units

Q19:

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

Q20:

Find the shortest distance between the line 𝑦 = 1 2 𝑥 − 2 and the point 𝐴 ( 9 , − 1 0 ) .

  • A √ 5
  • B √ 3
  • C √ 2 7 7
  • D 5 √ 5
  • E √ 1 1

Q21:

Determine, to the nearest hundredth, the distance between the point ( 7 , − 5 , − 4 ) , the straight line passing through the point ( 0 , − 2 , 2 ) , and its direction ratios ( − 9 , 7 , − 5 ) .

Q22:

Find the shortest distance between the point ( − 6 , 1 0 ) and the line which passes through the points ( 1 , 9 ) and ( 4 , 6 ) .

  • A √ 6
  • B √ 1 4
  • C 4 √ 2
  • D 3 √ 2
  • E 7 √ 2

Q23:

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

Q24:

Find the length of the perpendicular drawn from point 𝐴 ( − 8 , 1 , 1 0 ) to the straight line r = ⟨ − 1 , 2 , − 7 ⟩ + 𝑡 ⟨ − 9 , − 9 , 6 ⟩ rounded to the nearest hundredth.

Q25:

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 ) .

  • 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

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