Given a trapezoid $ABCD$ with its diagonals drawn and $E$ is the point where the diagonals intersect. Then the trapezoid is divided up into 4 triangles. Theres a well known theorem that if $k_{1},k_{2},k_{3},k_{4}$ are the areas of each of the four triangles then, $k_{1} \cdot k_{3} = k_{2} \cdot k_{4}$ Is it possible to find the area of the trapezoid if you only knew the area of two of the four triangles that make it up using the theorem above?
[Math] finding the area of a trapezoid using only 2 of the 4 triangles that makes up its interior
geometry
Best Answer
I have the figure drawn according to the given.
Also, by (2) again, $k_2 = k_4$ ….. (*)
If 2 of the areas of triangles are given, depending which two, we have to separate the study into the following cases. For simplicity, I will use {i} to represent $k_i$ for i = 1, … 4
Case-1 If {1} and {2} are known, then {4} is also known because of (*) and then {3} can be found from the given relation.
Case-2 If {1} and {4} are known, this same as case-1.
Case-3 If {2} and {4} are known, then $k_1 \cdot k_3 = (k_2)^2$. No further result can be developed.
Case-4 If {3} and {4} are known, similar to case-1.
Case-5 if {2} and {3} are known, similar to case-1.
Case-6 If {1} and {3} are known, then {2}= {4} $= \sqrt (k_1 \cdot k_3)$.