Worksheet: Distance between Two Points in Three Dimensions

In this worksheet, we will practice finding the coordinates of a point in three dimensions, the plane in which it is located, and the distance between two points.

Q1:

Find the distance between the two points 𝐴 ( 7 , 1 2 , 3 ) and 𝐵 ( 4 , 1 , 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:

Given that 𝐴 ( 𝑎 , 𝑏 , 𝑐 ) is the midpoint of the line segment between 𝐵 ( 9 , 1 7 , 2 ) and 𝐶 ( 1 6 , 1 2 , 7 ) , what is 𝑎 + 𝑏 + 𝑐 ?

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

Q3:

Given that 𝐶 1 2 , 0 , 2 is the midpoint of 𝐴 𝐵 , where the coordinates of 𝐴 and 𝐵 are ( 𝑘 + 5 , 8 , 𝑚 + 4 ) and ( 6 , 𝑛 + 7 , 5 ) , respectively, what is 𝑘 + 𝑚 𝑛 ?

Q4:

Given that point ( 5 𝑎 , 𝑎 + 2 , 1 4 ) lies in the 𝑥 𝑧 -plane, determine its distance from the 𝑦 𝑧 -plane.

Q5:

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 ( 1 , 0 , 1 ) .

Q6:

What is the distance between the point ( 1 9 , 5 , 5 ) and the 𝑥 -axis?

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

Q7:

The points 𝐴 , 𝐵 , and 𝐶 are on the 𝑥 -, 𝑦 -, and 𝑧 -axes, respectively. Given that ( 1 2 , 1 2 , 0 ) is the midpoint of 𝐴 𝐵 and ( 0 , 1 2 , 1 4 ) the midpoint of 𝐵 𝐶 , find the coordinates of the midpoint of 𝐴 𝐶 .

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

Q8:

Find 𝑘 so that the points ( 3 , 9 , 4 ) , ( 9 , 3 , 1 ) , ( 7 , 2 9 , 𝑘 ) are collinear.

Q9:

Given that point ( 4 𝑎 , 𝑎 + 1 0 , 9 ) lies in the 𝑥 𝑧 -plane, determine its distance from the 𝑦 𝑧 -plane.

Q10:

Given that point ( 6 𝑎 , 𝑎 + 4 , 1 4 ) lies in the 𝑥 𝑧 -plane, determine its distance from the 𝑦 𝑧 -plane.

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