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!