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.

  • A34
  • B34
  • C43
  • D43

Q2:

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

  • A0,32,10
  • B(0,3,20)
  • C(8,8,2)
  • D(16,13,4)

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(19,27,34)
  • B(19,24,4)
  • C(19,41,6)
  • D(19,10,24)

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(22,36,13)
  • B(5,15,20)
  • C(14,32,31)
  • D(13,19,2)

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(14,0,35)
  • B(11,12,19)
  • C(10,16,37)
  • D(13,4,17)

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.

  • A43
  • B43
  • C34
  • D34

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.

  • A56
  • B56
  • C65
  • D65

Q10:

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

  • A𝑦𝑧
  • B𝑥𝑦
  • C𝑥𝑧

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.

  • A𝐴(3,3,3)
  • B𝐴(3,3,4)
  • C𝐴(3,3,3)
  • D𝐴(0,0,4)

Q15:

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

  • A(0,12,7)
  • B(0,7,7)
  • C(0,5,7)
  • D(7,12,0)

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.

  • A𝐵(0,7,5), 𝐶(0,7,6)
  • B𝐵(0,0,6), 𝐶(0,5,0)
  • C𝐵(7,0,5), 𝐶(7,5,0)
  • D𝐵(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
  • B411 length units
  • C52 length units
  • D10 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(12,0,14)
  • C(24,0,28)
  • D(6,12,7)

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?

  • A32
  • B452
  • C172
  • D52

Q23:

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

  • A267length units
  • B299length units
  • C299length units
  • D267length 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.

  • A𝑘={0}
  • B𝑘={3}
  • C𝑘={4}
  • D𝑘={2}

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.