Geometry Area of sections of a parallelogram

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Homework Help Overview

The problem involves finding the area of four regions formed within a parallelogram when a segment is drawn from one vertex to the midpoint of an opposite side, and a diagonal is drawn between the other two vertices. The area of the parallelogram is given as 60 square units.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to analyze the areas of the regions formed by the segments and diagonal, noting that two triangles are similar with a ratio of 1/4. They express uncertainty about finding further relationships. Other participants suggest splitting one of the regions into smaller triangles to explore area relationships.

Discussion Status

Participants are actively engaging with the problem, offering hints and suggestions for further exploration. There is a recognition of the need to focus on area relationships rather than congruence. The original poster indicates progress in understanding, though the discussion remains open-ended.

Contextual Notes

The original poster has utilized Geometer Sketchpad to visualize the problem, which raises questions about how to approach the problem without such tools. There is an emphasis on understanding the relationships between the areas of the regions formed.

Wildcat
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1. The area of a parallelogram is 60 square units. A segment is drawn from one vertex to the midpoint of an opposite side. The diagonal is drawn between the other two other vertices. Find the area of the four regions formed.

Homework Equations





3. I found that two of the triangles formed are similar so the ratio of those triangles would be 1/4. The diagonal separates one region into two triangles which would total 30 units and the lower portion made up of a quadrilateral and a triangle would total 30 units. I'm stuck. I can't find another relationship. Help!

OK I constructed the diagram on geometer sketchpad and found the areas, but I'm wondering how it can be done without geometer sketchpad?? region I = 5 region II=10 region III = 20 and region IV (quadrilateral) = 25. Any ideas??
 
Last edited:
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Hi Wildcat! :smile:

Hint: split region IV into triangles V and VI, and find relations between I II III and V :wink:
 
tiny-tim said:
Hi Wildcat! :smile:

Hint: split region IV into triangles V and VI, and find relations between I II III and V :wink:

I did that. And I know from what I did on GSP that One of the triangles say V is congruent to II, but I can't figure out why. Can you give me another hint :)
 
forget congruence, think areas :wink:
 
tiny-tim said:
forget congruence, think areas :wink:

Thanks, I finally got it :)
 

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