2021 Fall AMC 12A Problems/Problem 21
Contents
Problem
Let be an isosceles trapezoid with and . Points and lie on diagonal with between and , as shown in the figure. Suppose , , , and . What is the area of
Solution 1
First realize that Thus, because we can say that and From the Pythagorean Theorem, we have and Because from the problem statement, we have that Solving, gives To find the area of the trapezoid, we can compute the area of and add it to the area of Thus, the area of the trapezoid is Thus, the answer is
~NH14
Solution 2
We put the graph to a coordinate plane. We put to the origin, to the -axis, and to the -axis.
We have the coordinates of the following points: , , , .
Denote , . Hence, , .
Hence, the slope of is . The slope of is .
Because , . Hence,
In , following from the Pythagorean theorem, we have .
In , following from the Pythagorean theorem, we have .
Because ,
Solving Equations (1) and (2), we get , .
Therefore,
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
Solution 3
Since both and are perpendicular to , we have .
Since the trapezoid is equilateral, . Hence, .
Notice that both triangles and share a common base .
Therefore, ~RunyangWang
Video Solution by The Power of Logic
~math2718281828459
Video Solution by Mathematical Dexterity
https://www.youtube.com/watch?v=TCeYkekkjrU
Video Solution by TheBeautyofMath
~IceMatrix
Video Solution by OmegaLearn
https://youtu.be/FWmrHV1dWPM?t=950
~ pi_is_3.14
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |