2021 AMC 12B Problems/Problem 17
Contents
Problem
Let be an isoceles trapezoid having parallel bases and with Line segments from a point inside to the vertices divide the trapezoid into four triangles whose areas are and starting with the triangle with base and moving clockwise as shown in the diagram below. What is the ratio
Solution
Without the loss of generality, let have vertices , , , and , with and . Also denote by the point in the interior of .
Let and be the feet of the perpendiculars from to and , respectively. Observe that and . Now using the formula for the area of a trapezoid yields Thus, the ratio satisfies ; solving yields .
(Observe that the given areas of and are irrelevant to the ratio .)
Solution 2
Let be the bottom base, be the top base, be the height of the bottom triangle, be the height of the top triangle. Thus, so Let so we get This gives us a quadratic in ie. so
- Solution by MathAwesome123, added by ccx09
Video Solution by OmegaLearn (Triangle Ratio and Trapezoid Area)
~ pi_is_3.14
See Also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.