2018 AMC 10 A
Complete problem set with solutions and individual problem pages
All of the triangles in the diagram below are similar to isosceles triangle , in which . Each of the smallest triangles has area , and has area . What is the area of trapezoid ? (2018 AMC 10A Problem, Question#9)
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Let be the area of . Note that is comprised of the small isosceles triangles and a triangle similar to with side length ratio (so an area ratio of ). Thus, we have .This gives , so the area of .
Let the base length of the small triangle be . Then, there is a triangle encompassing the small triangles and sharing the top angle with a base length of . Because the area is proportional to the square of the side, let the base be . Then triangle has an area of . So the area is .
Notice . Let the base of the small triangles of area be , then the base length of . Notice, ,
then Thus, .
The area of is times the area of the small triangle, as they are similar and their side ratio is . Therefore the area of the trapezoid is .
You can see that we can create a "stack" of triangles congruent to the small triangles shown here, arranged in a row above those , whose total area would be . Similarly, we can create another row of , and finally more at the top, as follows. We know this cumulative area will be , so to find the area of such trapezoid , we just take , like so.
The combined area of the small triangles is , and from the fact that each small triangle has an area of , we can deduce that the larger triangle above has an area of (as the sides of the triangles are in a proportion of , so will their areas have a proportion that is the square of the proportion of their sides, or ). Thus, the combined area of the top triangle and the trapezoid immediately below is . The area of trapezoid is thus the area of triangle .
