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\(ABCD\) is a regular tetrahedron (right triangular pyramid). If \(M\)is the midpoint of \(\overline{CD}\) ,then what is \(\tan\angle AMB\)?

 Feb 19, 2023
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Let's first draw a diagram of the regular tetrahedron ABCD and the point M, the midpoint of CD.

 

Since ABCD is a regular tetrahedron, we know that all of its edges are congruent and all of its faces are equilateral triangles. Let's call the length of an edge of ABCD "a".

 

]We can use trigonometry to find the tangent of the angle AMB. To do so, we need to find the length of AM and BM.

Since M is the midpoint of CD, we know that DM = MC = a/2.

We can use the Pythagorean theorem to find the length of AM:

AM^2 = AB^2 - BM^2

 

Since ABCD is a regular tetrahedron, we know that triangle ABC is an equilateral triangle with side length a. Therefore, we can use the Pythagorean theorem to find the length of AB:

 

AB^2 = AM^2 + BM^2 AB^2 = 4AM^2 (because AB = AC = BC = BD = CD = a)

Substituting the second equation into the first, we get:

AM^2 = 4AM^2 - BM^2 BM^2 = 3AM^2 BM = AM * sqrt(3)

 

Now that we know the lengths of AM and BM, we can find the tangent of angle AMB:

 

tan(AMB) = BM / AM tan(AMB) = (AM * sqrt(3)) / AM tan(AMB) = sqrt(3)

 

Therefore, the tangent of angle AMB is sqrt(3).

 Feb 19, 2023

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