Worksheet: Points, Midpoints, and Distances in Space

In this worksheet, we will practice finding the coordinates of a point in 3D, the distance between two points in 3D, and the coordinates of a midpoint and an endpoint in 3D using the formula.

Q1:

Given that the midpoint of 𝐴𝐵 lies in the 𝑥𝑦-plane, and the coordinates of 𝐴 and 𝐵 are (12,9,𝑘+3) and (15,9,3𝑘), respectively, determine the value of 𝑘.

  • A 3 4
  • B 3 4
  • C 4 3
  • D 4 3

Q2:

Points 𝐴, 𝐵 have coordinates (8,8,12) and (8,5,8), respectively. Determine the coordinates of the midpoint of 𝐴𝐵.

  • A 0 , 3 2 , 1 0
  • B ( 0 , 3 , 2 0 )
  • C ( 8 , 8 , 2 )
  • D ( 1 6 , 1 3 , 4 )

Q3:

Determine, to the nearest hundredth, the perimeter of the triangle formed by joining the midpoints of the sides of 𝐴𝐵𝐶, given that the coordinates of 𝐴, 𝐵, and 𝐶 are (10,8,2), (8,7,10), and (2,3,14), respectively.

Q4:

Given that point (0,17,10) is the midpoint of 𝐴𝐵 and that 𝐴(19,7,14), what are the coordinates of 𝐵?

  • A ( 1 9 , 2 7 , 3 4 )
  • B ( 1 9 , 2 4 , 4 )
  • C ( 1 9 , 4 1 , 6 )
  • D ( 1 9 , 1 0 , 2 4 )

Q5:

Given that point (9,17,11) is the midpoint of 𝐴𝐵 and that 𝐴(4,2,9), what are the coordinates of 𝐵?

  • A ( 2 2 , 3 6 , 1 3 )
  • B ( 5 , 1 5 , 2 0 )
  • C ( 1 4 , 3 2 , 3 1 )
  • D ( 1 3 , 1 9 , 2 )

Q6:

Given that point (1,4,18) is the midpoint of 𝐴𝐵 and that 𝐴(12,8,1), what are the coordinates of 𝐵?

  • A ( 1 4 , 0 , 3 5 )
  • B ( 1 1 , 1 2 , 1 9 )
  • C ( 1 0 , 1 6 , 3 7 )
  • D ( 1 3 , 4 , 1 7 )

Q7:

Determine, to the nearest hundredth, the perimeter of the triangle formed by joining the midpoints of the sides of 𝐴𝐵𝐶, given that the coordinates of 𝐴, 𝐵, and 𝐶 are (19,18,4), (1,4,16), and (13,18,3), respectively.

Q8:

Given that the midpoint of 𝐴𝐵 lies in the 𝑥𝑧-plane, and the coordinates of 𝐴 and 𝐵 are (14,𝑘+4,19) and (17,2𝑘,18), respectively, determine the value of 𝑘.

  • A 4 3
  • B 4 3
  • C 3 4
  • D 3 4

Q9:

Given that the midpoint of 𝐴𝐵 lies in the 𝑥𝑦-plane, and the coordinates of 𝐴 and 𝐵 are (3,18,𝑘+5) and (19,1,5𝑘), respectively, determine the value of 𝑘.

  • A 5 6
  • B 5 6
  • C 6 5
  • D 6 5

Q10:

In which of the following coordinate planes does the point (7,8,0) lie?

  • A 𝑦 𝑧
  • B 𝑥 𝑦
  • C 𝑥 𝑧

Q11:

Given that the point (𝑥,𝑦,𝑧) lies in the 𝑥𝑦-plane, determine its 𝑧-coordinate.

Q12:

Given that the point (𝑥,𝑦,𝑧) lies in the 𝑦𝑧-plane, determine its 𝑥-coordinate.

Q13:

Given that the point (𝑥,𝑦,𝑧) lies in the 𝑥𝑧-plane, determine its 𝑦-coordinate.

Q14:

Determine the coordinates of point 𝐴.

  • A 𝐴 ( 3 , 3 , 3 )
  • B 𝐴 ( 3 , 3 , 4 )
  • C 𝐴 ( 3 , 3 , 3 )
  • D 𝐴 ( 0 , 0 , 4 )

Q15:

Given that the point (7𝑚,12𝑚,𝑚) lies in the 𝑦𝑧-plane, what is this point?

  • A ( 0 , 1 2 , 7 )
  • B ( 0 , 7 , 7 )
  • C ( 0 , 5 , 7 )
  • D ( 7 , 1 2 , 0 )

Q16:

In the figure shown, the points 𝑂 and 𝐴 have coordinates (0,0,0) and (7,5,6), respectively. Determine the coordinates of 𝐵 and 𝐶.

  • A 𝐵 ( 0 , 7 , 5 ) , 𝐶 ( 0 , 7 , 6 )
  • B 𝐵 ( 0 , 0 , 6 ) , 𝐶 ( 0 , 5 , 0 )
  • C 𝐵 ( 7 , 0 , 5 ) , 𝐶 ( 7 , 5 , 0 )
  • D 𝐵 ( 7 , 5 , 0 ) , 𝐶 ( 7 , 0 , 6 )

Q17:

What is the distance between the point (19,5,5) and the 𝑥-axis?

  • A19 length units
  • B 4 1 1 length units
  • C 5 2 length units
  • D 1 0 length units

Q18:

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

Q19:

The points 𝐴, 𝐵, and 𝐶 are on the 𝑥-, 𝑦-, and 𝑧-axes, respectively. Given that (12,12,0) is the midpoint of 𝐴𝐵 and (0,12,14) the midpoint of 𝐵𝐶, find the coordinates of the midpoint of 𝐴𝐶.

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

Q20:

Given that 𝐶12,0,2 is the midpoint of 𝐴𝐵, where the coordinates of 𝐴 and 𝐵 are (𝑘+5,8,𝑚+4) and (6,𝑛+7,5), respectively, what is 𝑘+𝑚𝑛?

Q21:

Given that point (5𝑎,𝑎+2,14) lies in the 𝑥𝑧-plane, determine its distance from the 𝑦𝑧-plane.

Q22:

Given that 𝐴(𝑎,𝑏,𝑐) is the midpoint of the line segment between 𝐵(9,17,2) and 𝐶(16,12,7), what is 𝑎+𝑏+𝑐?

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

Q23:

Find the distance between the two points 𝐴(7,12,3) and 𝐵(4,1,8).

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

Q24:

Find 𝑘 so that the points (3,9,4), (9,3,1), (7,29,𝑘) are collinear.

Q25:

Given that the points (6,4,3), (7,6,𝑘), and (5,5,1) are the vertices of a triangle, determine all the possible values of 𝑘 that make the triangle equilateral.

  • A 𝑘 = { 0 }
  • B 𝑘 = { 3 }
  • C 𝑘 = { 4 }
  • D 𝑘 = { 2 }

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