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| ==Problem== | | ==Problem== |
− | Sides <math>AB</math>, <math>BC</math>, <math>CD</math>, and <math>DA</math>, respectively of convex quadrilateral <math>ABCD</math> are extended past <math>B</math>, <math>C</math>, <math>D</math>, and <math>A</math> to points <math>B'</math>, <math>C'</math>, <math>D'</math>, and <math>A'</math>. If <math>AB = BB' = 6</math>, <math>BC = CC' = 7</math>, <math>CD = DD' = 8</math>, and <math>DA = AA' = 9</math>, and the area of <math>ABCD</math> is 10, determine the area of quadrilateral <math>A'B'C'D'</math>. | + | Sides <math>AB,~ BC, ~CD</math> and <math>DA</math>, respectively, of convex quadrilateral <math>ABCD</math> are extended past <math>B,~ C ,~ D</math> and <math>A</math> to points <math>B',~C',~ D'</math> and <math>A'</math>. Also, <math>AB = BB' = 6,~ BC = CC' = 7, ~CD = DD' = 8</math> and <math>DA = AA' = 9</math>; and the area of <math>ABCD</math> is <math>10</math>. The area of <math>A 'B 'C'D'</math> is |
− | | |
− | <math></math>
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− | [asy]
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− | unitsize(1 cm);
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− | | |
− | pair[] A, B, C, D;
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− | A[0] = (0,0);
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− | B[0] = (0.6,1.2);
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− | C[0] = (-0.3,2.5);
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− | D[0] = (-1.5,0.7);
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− | B[1] = interp(A[0],B[0],2);
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− | C[1] = interp(B[0],C[0],2);
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− | D[1] = interp(C[0],D[0],2);
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− | A[1] = interp(D[0],A[0],2);
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− | | |
− | draw(A[1]--B[1]--C[1]--D[1]--cycle);
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− | draw(A[0]--B[1]);
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− | draw(B[0]--C[1]);
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− | draw(C[0]--D[1]);
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− | draw(D[0]--A[1]);
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− | | |
− | label("<math>A</math>", A[0], SW);
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− | label("<math>B</math>", B[0], SE);
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− | label("<math>C</math>", C[0], NE);
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− | label("<math>D</math>", D[0], NW);
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− | label("<math>A'</math>", A[1], SE);
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− | label("<math>B'</math>", B[1], NE);
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− | label("<math>C'</math>", C[1], N);
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− | label("<math>D'</math>", D[1], SW);
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− | | |
− | [\asy]
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− | <cmath>
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− | | |
− | </cmath>
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− | [asy]
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− | unitsize(1 cm);
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− | | |
− | pair[] A, B, C, D;
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− | | |
− | A[0] = (0,0);
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− | B[0] = (0.6,1.2);
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− | C[0] = (-0.3,2.5);
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− | D[0] = (-1.5,0.7);
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− | B[1] = interp(A[0],B[0],2);
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− | C[1] = interp(B[0],C[0],2);
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− | D[1] = interp(C[0],D[0],2);
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− | A[1] = interp(D[0],A[0],2);
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− | | |
− | draw(A[1]--B[1]--C[1]--D[1]--cycle);
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− | draw(A[0]--B[1]);
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− | draw(B[0]--C[1]);
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− | draw(C[0]--D[1]);
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− | draw(D[0]--A[1]);
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− | | |
− | label("<math>A</math>", A[0], SW);
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− | label("<math>B</math>", B[0], SE);
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− | label("<math>C</math>", C[0], NE);
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− | label("<math>D</math>", D[0], NW);
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− | label("<math>A'</math>", A[1], SE);
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− | label("<math>B'</math>", B[1], NE);
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− | label("<math>C'</math>", C[1], N);
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− | label("<math>D'</math>", D[1], SW);
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− | | |
− | [\asy]
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− | <cmath>
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− | | |
− | </cmath>
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− | [asy]
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− | unitsize(1 cm);
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− | | |
− | pair[] A, B, C, D;
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− | | |
− | A[0] = (0,0);
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− | B[0] = (0.6,1.2);
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− | C[0] = (-0.3,2.5);
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− | D[0] = (-1.5,0.7);
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− | B[1] = interp(A[0],B[0],2);
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− | C[1] = interp(B[0],C[0],2);
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− | D[1] = interp(C[0],D[0],2);
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− | A[1] = interp(D[0],A[0],2);
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− | | |
− | draw(A[1]--B[1]--C[1]--D[1]--cycle);
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− | draw(A[0]--B[1]);
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− | draw(B[0]--C[1]);
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− | draw(C[0]--D[1]);
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− | draw(D[0]--A[1]);
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− | | |
− | label("<math>A</math>", A[0], SW);
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− | label("<math>B</math>", B[0], SE);
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− | label("<math>C</math>", C[0], NE);
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− | label("<math>D</math>", D[0], NW);
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− | label("<math>A'</math>", A[1], SE);
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− | label("<math>B'</math>", B[1], NE);
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− | label("<math>C'</math>", C[1], N);
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− | label("<math>D'</math>", D[1], SW);
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− | | |
− | [\asy]
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− | <math></math>
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| ==Solution== | | ==Solution== |