Geometry Area of sections of a parallelogram

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SUMMARY

The area of a parallelogram is established as 60 square units. A segment drawn from one vertex to the midpoint of the opposite side, along with a diagonal connecting the other two vertices, divides the shape into four distinct regions. The areas of these regions are determined as follows: Region I = 5 square units, Region II = 10 square units, Region III = 20 square units, and Region IV (a quadrilateral) = 25 square units. The discussion highlights the use of Geometer Sketchpad for visualizing the problem, while also emphasizing the importance of understanding area relationships rather than congruence.

PREREQUISITES
  • Understanding of basic geometry concepts, specifically parallelograms
  • Knowledge of area calculation for triangles and quadrilaterals
  • Familiarity with geometric visualization tools, such as Geometer Sketchpad
  • Ability to analyze relationships between geometric figures
NEXT STEPS
  • Explore advanced geometric properties of parallelograms
  • Learn techniques for calculating areas of irregular shapes
  • Investigate the use of Geometer Sketchpad for geometric proofs
  • Study the principles of similarity and congruence in triangles
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Students studying geometry, educators teaching geometric concepts, and anyone interested in problem-solving techniques related to area calculations in polygons.

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.

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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??
 
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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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