1. [ASK] Pyramid in a Cube

Given a cube ABCD.EFGH whose side length is 4 cm. If the point I, K, and J is dividing EF, FG, and BG to two equal lengths respectively, determine the volume of pyramid D.IJK!

I think I can work out the pyramid's base area by deriving for the formula of equilateral triangle area. What I can't is determine the pyramid's height. All I knew is that is must be less than 4√3 cm. Anyone willing to help me?

2. Re: [ASK] Pyramid in a Cube

I don't know if it makes a difference but in my drawing
$ABCD$ is the top and $EFGH$ is the bottom of the cube

find the area of the right triangle $IJK$ (base of the pyramid)

Let $L$ and $M$ be the midpoints of $BC$ and $BA$ resp.

$DN$ is perpendicular to $LM$, $N$ on $LM$

It looks like $DN=$ the height of he pyramid

3. Re: [ASK] Pyramid in a Cube Originally Posted by Monoxdifly Given a cube ABCD.EFGH whose side length is 4 cm. If the point I, K, and J is dividing EF, FG, and BG to two equal lengths respectively, determine the volume of pyramid D.IJK!
I am sorry, the bold part was supposed to be BF. Must've been a typo.

4. Re: [ASK] Pyramid in a Cube

triangle $IJK$ is equilateral with side $=\sqrt 8$

the other three faces of the tetrahedron are congruent isosceles triangles

$DI=DJ=DK=6$

5. Re: [ASK] Pyramid in a Cube

The height, though? I can't determine the pyramid's height with that info because I'm not sure whether the center of the base is located in the middle point of the triangle base or not.

6. Re: [ASK] Pyramid in a Cube

Let $P$ be the foot of the perpendicular from $D$ onto the plane $IJK$

Let $L$ be the midpoint of $IK$

$\displaystyle \text{DL } \bot \text{ IK}$ since $DIK$ is isosceles

$\displaystyle \text{DP } \bot \text{ IK}$

plane $DPL$ perpendicular to line $IK$

Therefore $\displaystyle \text{PL } \bot \text{ IK}$

so $PL$ is the perpendicular bisector of side $IK$

similarly for the other two sides of triangle $IJK$

therefore $P$ is the center of gravity of triangle $IJK$

I get $DP=\frac{10}{\sqrt{3}}$ and volume $=\frac{20}{3}$

7. Re: [ASK] Pyramid in a Cube

Thank you Bro!