Area of a Quadrilateral within a Triangle

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Let be a triangle and let and be points on and , respectively. Let . You can calculate the area of the triangle by checking "coordinates" to find the base and height. You can drag the points and . Suppose that and that .

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What is ? See Details for the solution.

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Contributed by: Abraham Gadalla (November 2016)
Open content licensed under CC BY-NC-SA


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Details

This Demonstration changes the original problem, which stated that the areas are 3, 7 and 7, to express areas in terms of ratios.

Since is any point on segment , and the ratios of the areas of the two triangles and are equal, has to be the midpoint of .

Let the ratio to be determined be . Then the areas of the triangles , and are , and , respectively.

Since , then .

Hence also.

Let . Then:

It follows that , that is, . The area of the quadrilateral is .

Also, .

Therefore .

Reference

[1] Mathematics Competitions, 21(1), 2008. www.wfnmc.org/journal.html.



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