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Difficulty: MediumPythagorean Theorem and Special Right Triangles

A right rectangular prism has a square base. The diagonal of the base has a length of 626\sqrt{2} units. A space diagonal of the prism makes a 30∘30^\circ angle with the diagonal of the base. What is the height of the prism?

  1. 262\sqrt{6}Answer
  2. B
    666\sqrt{6}
  3. C
    626\sqrt{2}
  4. D
    12212\sqrt{2}
  5. E
    464\sqrt{6}

Answer

262\sqrt{6}
The height of a right rectangular prism is perpendicular to the base, forming a right triangle with the base diagonal as one leg and the space diagonal as the hypotenuse. Given that the angle between the space diagonal and the base diagonal is 30∘30^\circ, this right triangle is a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle. The base diagonal of length 626\sqrt{2} is adjacent to the 30∘30^\circ angle, making it the longer leg (x3x\sqrt{3}). Dividing 626\sqrt{2} by 3\sqrt{3} gives h=26h = 2\sqrt{6}, which correctly matches the length of the leg opposite the 30∘30^\circ angle.

Step-by-Step Solution

1
Identify the right triangle inside the prism.
The vertical right triangle has legs dd (base diagonal) and hh (height), with hypotenuse DD (space diagonal).
The height of a right prism is perpendicular to its base, forming a right angle with any line segment lying in the base, including the base diagonal.
2
Apply special right triangle ratios for a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle.
The base diagonal d=62d = 6\sqrt{2} is adjacent to the 30∘30^\circ angle, so d=x3d = x\sqrt{3}, where x=hx = h is the height opposite the 30∘30^\circ angle.
In a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle, side lengths follow the ratio 1:3:21 : \sqrt{3} : 2.
3
Solve for height hh.
h=623=663=26h = \frac{6\sqrt{2}}{\sqrt{3}} = \frac{6\sqrt{6}}{3} = 2\sqrt{6}.
Rationalize the denominator by multiplying top and bottom by 3\sqrt{3}.

Key Concept

Special Right Triangles (30∘−60∘−90∘30^\circ-60^\circ-90^\circ) in 3D Space Diagonals
Estimated Time:1m 30s
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Topics

Pythagorean Theorem and Special Right Triangles

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