Difference between revisions of "2021 Fall AMC 10B Problems/Problem 13"
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import olympiad; | import olympiad; | ||
pair A,B,C,D,E,F,G,H,I,J,K; | pair A,B,C,D,E,F,G,H,I,J,K; | ||
− | A = origin; B = (0. | + | A = origin; B = (0.5,0); C=(2.5,0); D=(3,0); E = (0.5,2); F=(0.83333333333,2); G=(2.166666666667,2); H=(2.5,2); I=(0.83333333333,3.333333333333); J=(2.166666666667,3.3333333333); K=(1.5,6); |
draw(A--D--K--cycle); | draw(A--D--K--cycle); | ||
draw(B--E); | draw(B--E); | ||
Line 39: | Line 39: | ||
draw(I--J); | draw(I--J); | ||
draw(E--H); | draw(E--H); | ||
− | draw(K--( | + | draw(K--(1.5,0)); |
− | label("$A$", | + | label("$A$",(1.5,0),S); |
− | |||
label("$C$",C,S); | label("$C$",C,S); | ||
label("$D$",D,E); | label("$D$",D,E); | ||
− | |||
− | |||
label("$G$",G,S); | label("$G$",G,S); | ||
label("$H$",H,E); | label("$H$",H,E); | ||
− | |||
label("$J$",J,E); | label("$J$",J,E); | ||
label("$K$",K,N); | label("$K$",K,N); |
Revision as of 15:43, 24 November 2021
Contents
[hide]Problem
A square with side length is inscribed in an isosceles triangle with one side of the square along the base of the triangle. A square with side length has two vertices on the other square and the other two on sides of the triangle, as shown. What is the area of the triangle?
Solution 1
Let's split the triangle down the middle and label it:
Solution 2
By similarity, the height is and the base is . Thus the area is , or .
~Hefei417, or 陆畅 Sunny from China
See Also
2021 Fall AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.