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

Video

15:16

Sample Question Videos

Worksheet • 25 Questions • 1 Video

Q1:

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

  • A | 𝑦 | 1
  • B | 𝑦 | | π‘₯ | 1 1
  • C | π‘₯ | 1
  • D  | π‘₯ | + | 𝑦 | 1 2 1 2
  • E0

Q2:

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.

Q3:

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

Q4:

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

Q5:

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 3 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 6 9 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 8 √ 1 0 5 units length
  • B √ 1 0 1 6 units length
  • C 1 1 √ 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’ 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 = 2 5 2
  • 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 = 3 0

Q8:

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 .

  • A9 or 1 9
  • B βˆ’ 9 or βˆ’ 1 9
  • C18 or 2 9
  • D βˆ’ 3 6 or 3

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

Q10:

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

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

Q11:

Find the perpendicular distance between the point 𝐴 ( 2 , 2 0 ) and the π‘₯ -axis.

Q12:

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

  • A57 or 9
  • B83 or βˆ’ 6 1
  • C βˆ’ 5 7 or βˆ’ 9
  • D83 or βˆ’ 9
  • E βˆ’ 9 or βˆ’ 6 1

Q13:

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

  • A2
  • B 2 √ 2
  • C 4 √ 2
  • D4

Q14:

Let 𝐿 be the line through point ( 7 , 5 , 5 ) in the direction of vector ( 2 , 4 , βˆ’ 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 ( βˆ’ 6 , 8 , 9 ) that makes equal angles with the three coordinate axes. What is the distance between the point ( βˆ’ 4 , 5 , 3 ) and 𝐿 , to the nearest hundredth.

Q16:

Determine, to the nearest hundredth, the length of the perpendicular drawn from the point ( βˆ’ 5 , βˆ’ 7 , βˆ’ 1 0 ) to the straight line π‘₯ + 8 2 = 𝑦 βˆ’ 9 8 = 𝑧 + 7 βˆ’ 8 .

Q17:

What is the distance between lines ( βˆ’ 1 6 , βˆ’ 1 6 ) + π‘˜ ( 2 , 4 ) and ( 1 9 , βˆ’ 1 7 ) + π‘˜ ( 7 , 1 4 ) ?

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

Q18:

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

  • A 4 √ 4 1 length units
  • B20 length units
  • C6 length units
  • D 4 √ 3 4 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 π‘₯ βˆ’ 2 and the point 𝐴 ( 9 , βˆ’ 1 0 ) .

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

Q21:

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

Q22:

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

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

Q23:

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

Q24:

Find the length of the perpendicular drawn from point 𝐴 ( βˆ’ 8 , 1 , 1 0 ) to the straight line ⃑ π‘Ÿ = ( βˆ’ 1 , 2 , βˆ’ 7 ) + 𝑑 ( βˆ’ 9 , βˆ’ 9 , 6 ) rounded to the nearest hundredth.

Q25:

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