Chapter 10: Q31E (page 468)

The vertices of a tetrahedron are four vertices of a cube symmetrically chosen so that no two are adjacent. Show that the angle between the vertices of a tetrahedron is $109 {.5}^{\circ}$ .

Short Answer

Expert verified

Proved that Angle between Vertices of a tetrahedron is ${109.5}^{0}$ .

Step by step solution

Definition of tetrahedron

A tetrahedron also referred to as a triangle pyramid in geometry, is a polyhedron with four triangular faces, six straight edges, and four vertex corners.

Step-2: Basic use of trigonometry and Pythagoras Theorem.

In triangle AOD, assuming the Cube has two-unit-length-sides.

$\begin{matrix} OD = 1 \\ AD = \sqrt{1^{2} + 1^{2}} \\ = \sqrt{2} \\ AO = \sqrt{{AD}^{2} + {OD}^{2}} \\ = \sqrt{{(\sqrt{2})}^{2} + 1^{2}} \\ = \sqrt{3} \end{matrix}$

Using Trigonometry,

$\begin{matrix} Cos ∠ AOD = \frac{1}{\sqrt{3}} \\ = \frac{\sqrt{3}}{3} \end{matrix}$

$\begin{matrix} ∠ AOD = \cos^{- 1} \frac{\sqrt{3}}{3} \\ = 54 {.74}^{0} \end{matrix}$

So, the Required Angle

$\begin{matrix} ∠ AOB = 2 ∠ AOD \\ = 2 \times 54 {.74}^{0} \\ = 109 {.5}^{0} \end{matrix}$

Hence it is proved that Angle between Vertices of a tetrahedron is ${109.5}^{0}$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

The vertices of a tetrahedron are four vertices of a cube symmetrically chosen so that no two are adjacent. Show that the angle between the vertices of a tetrahedron is $109 {.5}^{\circ}$ .

Short Answer

Step by step solution

Definition of tetrahedron

Step-2: Basic use of trigonometry and Pythagoras Theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Mechanics and Materials

Electromagnetism

Kinematics Physics

Modern Physics

Translational Dynamics

Study anywhere. Anytime. Across all devices.

The vertices of a tetrahedron are four vertices of a cube symmetrically chosen so that no two are adjacent. Show that the angle between the vertices of a tetrahedron is 109.5∘ .

Short Answer

Step by step solution

Definition of tetrahedron

Step-2: Basic use of trigonometry and Pythagoras Theorem.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

Mechanics and Materials

Electromagnetism

Kinematics Physics

Modern Physics

Translational Dynamics

Study anywhere. Anytime. Across all devices.

The vertices of a tetrahedron are four vertices of a cube symmetrically chosen so that no two are adjacent. Show that the angle between the vertices of a tetrahedron is $109 {.5}^{\circ}$ .