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dazzle1230

Member

Registered: 2016-05-17

Posts: 92

regular tetrahedron problem

Let ABCD be a regular tetrahedron. (A regular tetrahedron is a pyramid in which all four faces are equilateral triangles.) The centers of the faces of tetrahedron ABCD are connected to form another regular tetrahedron EFGH. Let V1 and V2 be the volumes of tetrahedra ABCD and EFGH, respectively. Find V1/V2.

I know the volume of a regular tetrahedron is s^3/6sqrt(2) where s is the edge length.  But I don't know how to find the edge length of EFGH.

dazzle1230

Member

Registered: 2016-05-17

Posts: 92

Re: regular tetrahedron problem

Also, can I have help on this problem too:

Let ABCD be a regular tetrahedron and let P be the unique point equidistant from points A,B,C,D. Extend AP to hit face BCD at point Q. What is the ratio PQ/AQ?

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: regular tetrahedron problem

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

Bob

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Registered: 2010-06-20

Posts: 10,621

Re: regular tetrahedron problem

hi dazzle1230

I've distorted the diagram slightly so that points don't end up looking like they are on top of each other.

Original tetrahedron ABCD.  New tetrahedron EFGH. 

I, J and K are the midpoints of BC, CD and BD.

So DI, AJ, AI and AK are medians.  The centres are 2/3 of the way down from the top of the median.

Consider triangle IJK.  Because it is formed from the midpoints it will be similar to DBC and half its size.

Triangle HFG is also similar to IJK  and 2/3 of its size.

So you can work out the ratio BC:GF.  The volume ratio will be the cube of the length ratio.

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
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dazzle1230

Member

Registered: 2016-05-17

Posts: 92

Re: regular tetrahedron problem

Thanks so much!
How about the second question?

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: regular tetrahedron problem

---

{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: regular tetrahedron problem

Last edited by thickhead (2016-06-18 16:20:06)

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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}