i am confused in one problem and please help me,i will give picture from problem below
and question says : $AB$ is a diameter of the circle. All triangles above the diameter in the diagram are equal in area. All triangles below the diameter are equal in area.compare total area of triangles above $AB$ and total are below $AB$
i have chosen that this can't be determined by given information,because in spite of we have fact that below $AB$ we have more triangles then above $AB$,we dont know length of bases of each triangles,all sides except bases are radius so equal to each other,but in question answer is different,and explanation is following:
The total area of the upper triangles is less than the area of the lower triangles. The more triangles that you cut the semicircle into, the more of the circle that is occupied.
is this right?it is test taken from GRE test,i am preparing for passing it
Best Answer
GRE is correct; they in fact check if you know a famous proof of formula for circle area. The proof takes area of circle as limit of total area of triangles, which become smaller and smaller and closer to the circle, making a circle as a limit. Full proof is for example here, visualized: http://faraday.physics.utoronto.ca/GeneralInterest/Harrison/Flash/AreaOfCircle/AreaOfCircle.html
So just by taking more triangles, they will necessarily be closer to circumference and hence their total area is larger. GRE is about intuitiveness, so this reasoning is mainly intuitive but based on some knowledge you need to have.