Area of a Quadrilateral within a Triangle
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 .[more]
What is ? See Details for the solution.[less]
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 .
Let . Then:
It follows that , that is, . The area of the quadrilateral is .
 Mathematics Competitions, 21(1), 2008. www.wfnmc.org/journal.html.