A right rectangular prism has a square base. The diagonal of the base has a length of 6\sq… — GRE General Test

A right rectangular prism has a square base. The diagonal of the base has a length of $6\sqrt{2}$ units. A space diagonal of the prism makes a $30^\circ$ angle with the diagonal of the base. What is the height of the prism?

$2\sqrt{6}$ Cevap
B
$6\sqrt{6}$
C
$6\sqrt{2}$
D
$12\sqrt{2}$
E
$4\sqrt{6}$

Cevap

2\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^\circ

, this right triangle is a

30^\circ-60^\circ-90^\circ

triangle. The base diagonal of length

6\sqrt{2}

is adjacent to the

30^\circ

angle, making it the longer leg (

x\sqrt{3}

). Dividing

6\sqrt{2}

\sqrt{3}

gives

h = 2\sqrt{6}

, which correctly matches the length of the leg opposite the

30^\circ

angle.

Adım Adım Çözüm

Identify the right triangle inside the prism.

The vertical right triangle has legs

d

(base diagonal) and

h

(height), with hypotenuse

D

(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.

Apply special right triangle ratios for a

30^\circ-60^\circ-90^\circ

triangle.

The base diagonal

d = 6\sqrt{2}

is adjacent to the

30^\circ

angle, so

d = x\sqrt{3}

, where

x = h

is the height opposite the

30^\circ

angle.

In a

30^\circ-60^\circ-90^\circ

triangle, side lengths follow the ratio

1 : \sqrt{3} : 2

Solve for height

h

h = \frac{6\sqrt{2}}{\sqrt{3}} = \frac{6\sqrt{6}}{3} = 2\sqrt{6}

Rationalize the denominator by multiplying top and bottom by

\sqrt{3}

Anahtar Kavram

Special Right Triangles (

30^\circ-60^\circ-90^\circ

) in 3D Space Diagonals

Tahmini Süre:1m 30s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru