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Worksheet: Distance between Two Points in Three Dimensions

Q1:

Find the distance between the two points 𝐴 ( βˆ’ 7 , 1 2 , 3 ) and 𝐡 ( βˆ’ 4 , βˆ’ 1 , βˆ’ 8 ) .

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

Q2:

What is the distance between the point ( 1 9 , 5 , 5 ) and the π‘₯ -axis?

  • A √ 4 1 1 length units
  • B19 length units
  • C √ 1 0 length units
  • D 5 √ 2 length units

Q3:

Calculate, to two decimal places, the area of the triangle 𝑃 𝑄 𝑅 , where the coordinates of its vertices are at 𝑃 ( 4 , 0 , 2 ) , 𝑄 ( 2 , 1 , 5 ) , and 𝑅 ( βˆ’ 1 , 0 , 1 ) .

Q4:

The points 𝐴 , 𝐡 , and 𝐢 are on the π‘₯ -, 𝑦 -, and 𝑧 -axes, respectively. Given that ( 1 2 , βˆ’ 1 2 , 0 ) is the midpoint of 𝐴 𝐡 and ( 0 , βˆ’ 1 2 , βˆ’ 1 4 ) the midpoint of 𝐡 𝐢 , find the coordinates of the midpoint of 𝐴 𝐢 .

  • A ( 6 , 0 , 7 )
  • B ( 6 , βˆ’ 1 2 , βˆ’ 7 )
  • C ( 2 4 , 0 , βˆ’ 2 8 )
  • D ( 1 2 , 0 , βˆ’ 1 4 )

Q5:

Given that 𝐢 ο€Ό βˆ’ 1 2 , 0 , βˆ’ 2  is the midpoint of 𝐴 𝐡 , where the coordinates of 𝐴 and 𝐡 are ( π‘˜ + 5 , 8 , π‘š + 4 ) and ( βˆ’ 6 , 𝑛 + 7 , 5 ) , respectively, what is π‘˜ + π‘š βˆ’ 𝑛 ?

Q6:

Given that point ( 5 π‘Ž , π‘Ž + 2 , βˆ’ 1 4 ) lies in the π‘₯ 𝑧 -plane, determine its distance from the 𝑦 𝑧 -plane.

Q7:

Given that point ( π‘Ž + 3 , 4 π‘Ž , 1 9 ) lies in the 𝑦 𝑧 -plane, determine its distance from the π‘₯ 𝑧 -plane.

Q8:

Given that point ( 2 , 6 π‘Ž , π‘Ž + 3 ) lies in the π‘₯ 𝑦 -plane, determine its distance from the π‘₯ 𝑧 -plane.

Q9:

Given that 𝐴 ( π‘Ž , 𝑏 , 𝑐 ) is the midpoint of the line segment between 𝐡 ( 9 , βˆ’ 1 7 , 2 ) and 𝐢 ( 1 6 , βˆ’ 1 2 , 7 ) , what is π‘Ž + 𝑏 + 𝑐 ?

  • A 4 5 2
  • B βˆ’ 1 7 2
  • C 3 2
  • D 5 2

Q10:

Find π‘˜ so that the points ( 3 , 9 , βˆ’ 4 ) , ( 9 , βˆ’ 3 , βˆ’ 1 ) , ( βˆ’ 7 , 2 9 , π‘˜ ) are collinear.