Proving Ratio of Areas/Heights of Similar Pyramids

[IniquiTrance]

Homework Statement

If I have similar pyramids each with similar triangular bases, how can I prove that the ratio of the area of the bases = the ratio of the heights of the pyramids squared?

Thanks!

Homework Equations

The Attempt at a Solution

 

[Dick]

If the pyramids are similar then the ratio of the heights is the same as the ratio of the sides of the base triangle. If you change all of the sides of the triangle by a constant ratio, how does the area change?

 

[IniquiTrance]

I'm not sure how to proceed...

 

[IniquiTrance]

I see that if the scaling factor is k, then the ratio of the areas is k². But how does that relate to the height of the pyramids?

 

[IniquiTrance]

Hmm, I think I see it.

k equals the ratio of the sides of the bases which = the ratio of the heights. Squaring this equals the ratio of the areas.

Thanks!