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Triangles are formed in a cube by connecting three vertices.

A cube. The top face has points G, B, C, F and the bottom face has points H, A, D, E.

Which triangles are right triangles? Check all that apply.
triangle AEG
triangle AEH
triangle DAE
triangle BFH
triangle GDH
triangle CFG
triangle DFH
triangle BCF

1 answer

To determine which triangles among the given options are right triangles, we need to analyze the vertices of the cube in 3D space.

Let's label the vertices of the cube as follows with respect to coordinates:

  • Top face:

    • G (0, 0, 1)
    • B (1, 0, 1)
    • C (1, 1, 1)
    • F (0, 1, 1)

  • Bottom face:

    • H (0, 0, 0)
    • A (1, 0, 0)
    • D (1, 1, 0)
    • E (0, 1, 0)

Now, we can analyze each triangle to check if any of them are right triangles (a right triangle has one angle that is 90 degrees, which occurs when two sides are perpendicular).

  1. Triangle AEG:

    • Vectors:
      • AE = E - A = (0, 1, 0) - (1, 0, 0) = (-1, 1, 0)
      • AG = G - A = (0, 0, 1) - (1, 0, 0) = (-1, 0, 1)
    • Check dot product AE · AG = 0:
      • (-1, 1, 0) · (-1, 0, 1) = 1 (not perpendicular).
    • Not a right triangle.

  2. Triangle AEH:

    • Vectors:
      • AE = E - A = (-1, 1, 0)
      • AH = H - A = (0, 0, 0) - (1, 0, 0) = (-1, 0, 0)
    • Check dot product AE · AH = 0:
      • (-1, 1, 0) · (-1, 0, 0) = 1 (not perpendicular).
    • Not a right triangle.

  3. Triangle DAE:

    • Vectors:
      • DA = A - D = (1, 0, 0) - (1, 1, 0) = (0, -1, 0)
      • DE = E - D = (0, 1, 0) - (1, 1, 0) = (-1, 0, 0)
    • Check dot product DA · DE = 0:
      • (0, -1, 0) · (-1, 0, 0) = 0 (perpendicular).
    • This is a right triangle.

  4. Triangle BFH:

    • Vectors:
      • BF = F - B = (0, 1, 1) - (1, 0, 1) = (-1, 1, 0)
      • BH = H - B = (0, 0, 0) - (1, 0, 1) = (-1, 0, -1)
    • Check dot product BF · BH = 0:
      • (-1, 1, 0) · (-1, 0, -1) = 1 (not perpendicular).
    • Not a right triangle.

  5. Triangle GDH:

    • Vectors:
      • GD = D - G = (1, 1, 0) - (0, 0, 1) = (1, 1, -1)
      • GH = H - G = (0, 0, 0) - (0, 0, 1) = (0, 0, -1)
    • Check dot product GD · GH = 0:
      • (1, 1, -1) · (0, 0, -1) = -1 (not perpendicular).
    • Not a right triangle.

  6. Triangle CFG:

    • Vectors:
      • CF = F - C = (0, 1, 1) - (1, 1, 1) = (-1, 0, 0)
      • CG = G - C = (0, 0, 1) - (1, 1, 1) = (-1, -1, 0)
    • Check dot product CF · CG = 0:
      • (-1, 0, 0) · (-1, -1, 0) = 1 (not perpendicular).
    • Not a right triangle.

  7. Triangle DFH:

    • Vectors:
      • DF = F - D = (0, 1, 1) - (1, 1, 0) = (-1, 0, 1)
      • DH = H - D = (0, 0, 0) - (1, 1, 0) = (-1, -1, 0)
    • Check dot product DF · DH = 0:
      • (-1, 0, 1) · (-1, -1, 0) = 1 (not perpendicular).
    • Not a right triangle.

  8. Triangle BCF:

    • Vectors:
      • BC = C - B = (1, 1, 1) - (1, 0, 1) = (0, 1, 0)
      • BF = F - B = (0, 1, 1) - (1, 0, 1) = (-1, 1, 0)
    • Check dot product BC · BF = 0:
      • (0, 1, 0) · (-1, 1, 0) = 1 (not perpendicular).
    • Not a right triangle.

After evaluating all these triangles, the only triangle that is a right triangle is:

  • Triangle DAE.