Proving Ratio of Areas/Heights of Similar Pyramids

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Homework Help Overview

The discussion revolves around proving a relationship between the areas of the bases and the heights of similar pyramids, specifically focusing on the claim that the ratio of the area of the bases equals the ratio of the heights squared.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the relationship between the heights and the sides of the triangular bases of similar pyramids, questioning how changes in side lengths affect area. There is also a consideration of the scaling factor and its implications for both area and height.

Discussion Status

Some participants have offered insights regarding the relationship between the scaling factor and the areas, while others express uncertainty about the connections being drawn. The discussion appears to be moving towards understanding the implications of the scaling factor on both areas and heights.

Contextual Notes

Participants are navigating the definitions and relationships inherent in the properties of similar figures, particularly focusing on the implications of similarity in geometric shapes.

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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

 
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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?
 
I'm not sure how to proceed... :blushing:
 
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?
 
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! :cool:
 

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