General Chemistry 111
11/28/11
Chapter 9 Extra Credit Quiz
Name: _____________________________ Section #______________________
Tetrahedral geometry has a bond angle of about 109.5 °. Given that a tetrahedron is made up of 4 equilateral triangles each of length x, what is the exact distance in units of x from the exact center of the tetrahedron to one of the vertices? What is the exact value of the angle formed between the center and any two vertices? (Hint: From the center of an equilateral triangle to the corners, you can draw three lines that forms 3 triangles each with 1/3 the area of the original triangle. Something similar happens for the four pyrimids that can be formed from the center of the tetrahedron to each vertex.).
To find the exact distance in units of x from the exact center of the tetrahedron to one of the vertices, we can use the properties of equilateral triangles.
First, let's consider one of the equilateral triangles that make up the tetrahedron. The length of each side of the equilateral triangle is x.
Now, let's draw a line from the center of the equilateral triangle to one of its vertices. This line will pass through the midpoint of one of the sides of the equilateral triangle.
Using the hint provided, we know that from the center to the corners of the equilateral triangle, we can draw three lines that form three smaller triangles, each with 1/3 the area of the original triangle.
Therefore, the smaller triangles formed by drawing the lines from the center to each vertex have an area of (1/3) * (1/2) * x * (x * sqrt(3)) = (1/6) * x^2 * sqrt(3).
Since these smaller triangles are congruent, we can consider one of them. It has a base of x and a height of (x * sqrt(3))/2.
Using the formula for the area of a triangle (area = (1/2) * base * height), we can set up the equation:
(1/6) * x^2 * sqrt(3) = (1/2) * x * [(x * sqrt(3))/2].
Simplifying this equation, we get:
(1/6) * x^2 * sqrt(3) = (1/4) * x^2 * sqrt(3).
If we multiply both sides by 6, we can eliminate the denominators:
x^2 * sqrt(3) = (3/2) * x^2 * sqrt(3).
Now, we can divide both sides by x^2 * sqrt(3) to solve for x:
1 = 3/2.
This is clearly not true, and we have arrived at a contradiction.
Therefore, it is not possible to find the exact distance in units of x from the exact center of the tetrahedron to one of the vertices using the information given.
As for the exact value of the angle formed between the center and any two vertices, it can be found by considering the four pyramids that can be formed from the center of the tetrahedron to each vertex.
Each of these pyramids can be divided into two congruent triangular pyramids, with one of the equilateral triangles of the tetrahedron as their base and the center of the tetrahedron at their apex.
Since the tetrahedron has four equilateral triangles, we have four congruent triangular pyramids.
The angle formed between the center of the tetrahedron and any two vertices is the same as the angle formed between the center of the tetrahedron and the apex of one of the triangular pyramids.
Since each of the four triangular pyramids is congruent, the angle formed is the same for all of them. However, without additional information given, we are unable to determine the exact value of this angle.