Difference between revisions of "1989 AHSME Problems/Problem 21"
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A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up 36% of the area of the flag, what percent of the area of the flag is blue? | A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up 36% of the area of the flag, what percent of the area of the flag is blue? | ||
| − | + | [asy] | |
| − | + | unitsize(2.5 cm); | |
| − | + | ||
| − | + | pair[] A, B, C; | |
| − | + | real t = 0.2; | |
| − | + | ||
| − | + | A[1] = (0,0); | |
| − | + | A[2] = (1,0); | |
| − | + | A[3] = (1,1); | |
| − | + | A[4] = (0,1); | |
| − | + | B[1] = (t,0); | |
| − | + | B[2] = (1 - t,0); | |
| + | B[3] = (1,t); | ||
| + | B[4] = (1,1 - t); | ||
| + | B[5] = (1 - t,1); | ||
| + | B[6] = (t,1); | ||
| + | B[7] = (0,1 - t); | ||
| + | B[8] = (0,t); | ||
| + | C[1] = extension(B[1],B[4],B[7],B[2]); | ||
| + | C[2] = extension(B[3],B[6],B[1],B[4]); | ||
| + | C[3] = extension(B[5],B[8],B[3],B[6]); | ||
| + | C[4] = extension(B[7],B[2],B[5],B[8]); | ||
| + | |||
| + | fill(C[1]--C[2]--C[3]--C[4]--cycle,blue); | ||
| + | fill(A[1]--B[1]--C[1]--C[4]--B[8]--cycle,red); | ||
| + | fill(A[2]--B[3]--C[2]--C[1]--B[2]--cycle,red); | ||
| + | fill(A[3]--B[5]--C[3]--C[2]--B[4]--cycle,red); | ||
| + | fill(A[4]--B[7]--C[4]--C[3]--B[6]--cycle,red); | ||
| + | |||
| + | draw(A[1]--A[2]--A[3]--A[4]--cycle); | ||
| + | draw(B[1]--B[4]); | ||
| + | draw(B[2]--B[7]); | ||
| + | draw(B[3]--B[6]); | ||
| + | draw(B[5]--B[8]); | ||
| + | [/asy] | ||
<math>\text{(A)}\ 0.5\qquad\text{(B)}\ 1\qquad\text{(C)}\ 2\qquad\text{(D)}\ 3\qquad\text{(E)}\ 6</math> | <math>\text{(A)}\ 0.5\qquad\text{(B)}\ 1\qquad\text{(C)}\ 2\qquad\text{(D)}\ 3\qquad\text{(E)}\ 6</math> | ||
Revision as of 09:24, 27 March 2020
Problem
A square flag has a red cross of uniform width with a blue square in the center on a white background as shown. (The cross is symmetric with respect to each of the diagonals of the square.) If the entire cross (both the red arms and the blue center) takes up 36% of the area of the flag, what percent of the area of the flag is blue?
[asy] unitsize(2.5 cm);
pair[] A, B, C; real t = 0.2;
A[1] = (0,0); A[2] = (1,0); A[3] = (1,1); A[4] = (0,1); B[1] = (t,0); B[2] = (1 - t,0); B[3] = (1,t); B[4] = (1,1 - t); B[5] = (1 - t,1); B[6] = (t,1); B[7] = (0,1 - t); B[8] = (0,t); C[1] = extension(B[1],B[4],B[7],B[2]); C[2] = extension(B[3],B[6],B[1],B[4]); C[3] = extension(B[5],B[8],B[3],B[6]); C[4] = extension(B[7],B[2],B[5],B[8]);
fill(C[1]--C[2]--C[3]--C[4]--cycle,blue); fill(A[1]--B[1]--C[1]--C[4]--B[8]--cycle,red); fill(A[2]--B[3]--C[2]--C[1]--B[2]--cycle,red); fill(A[3]--B[5]--C[3]--C[2]--B[4]--cycle,red); fill(A[4]--B[7]--C[4]--C[3]--B[6]--cycle,red);
draw(A[1]--A[2]--A[3]--A[4]--cycle); draw(B[1]--B[4]); draw(B[2]--B[7]); draw(B[3]--B[6]); draw(B[5]--B[8]); [/asy]
Solution
The diagram can be quartered as shown:
![[asy] draw((0,0)--(0,5)--(5,5)--(5,0)--(0,0)); draw((0,1)--(4,5)); draw((1,0)--(5,4)); draw((0,4)--(4,0)); draw((1,5)--(5,1)); draw((0,0)--(5,5),dotted); draw((0,5)--(5,0),dotted); [/asy]](http://latex.artofproblemsolving.com/4/a/7/4a7bcd7a4570fa60ff6e1cb35a3022bef95b5892.png)
and reassembled into two smaller squares of side 
, each of which looks like this:
![[asy] draw((0,0)--(0,5)--(5,5)--(5,0)--(0,0)); draw((0,1)--(4,1)--(4,5)); draw((1,0)--(1,4)--(5,4)); label("blue",(0.5,0.5)); label("blue",(4.5,4.5)); label("red",(0.5,4.5)); label("red",(4.5,0.5)); label("white",(2.5,2.5)); [/asy]](http://latex.artofproblemsolving.com/3/f/8/3f820cdb6c67538a1a453e16902ae8ead1ab3c4f.png)
The border in this figure is the former cross, which still occupies 36% of the area. Therefore the inner square occupies 64% of the area, from which we deduce that it is 
, and that one blue square must be 
or 1% each. Thus the blue area is 
of the total.
See also
| 1989 AHSME (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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
