Difference between revisions of "1978 AHSME Problems/Problem 29"
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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</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>. | ||
| + | |||
| + | <math></math> | ||
| + | [asy] | ||
| + | unitsize(1 cm); | ||
| + | |||
| + | pair[] A, B, C, D; | ||
| + | |||
| + | A[0] = (0,0); | ||
| + | B[0] = (0.6,1.2); | ||
| + | C[0] = (-0.3,2.5); | ||
| + | D[0] = (-1.5,0.7); | ||
| + | B[1] = interp(A[0],B[0],2); | ||
| + | C[1] = interp(B[0],C[0],2); | ||
| + | D[1] = interp(C[0],D[0],2); | ||
| + | A[1] = interp(D[0],A[0],2); | ||
| + | |||
| + | draw(A[1]--B[1]--C[1]--D[1]--cycle); | ||
| + | draw(A[0]--B[1]); | ||
| + | draw(B[0]--C[1]); | ||
| + | draw(C[0]--D[1]); | ||
| + | draw(D[0]--A[1]); | ||
| + | |||
| + | label("<math>A</math>", A[0], SW); | ||
| + | label("<math>B</math>", B[0], SE); | ||
| + | label("<math>C</math>", C[0], NE); | ||
| + | label("<math>D</math>", D[0], NW); | ||
| + | label("<math>A'</math>", A[1], SE); | ||
| + | label("<math>B'</math>", B[1], NE); | ||
| + | label("<math>C'</math>", C[1], N); | ||
| + | label("<math>D'</math>", D[1], SW); | ||
| + | |||
| + | [\asy] | ||
| + | <math></math> | ||
==Solution== | ==Solution== | ||
Revision as of 13:19, 23 August 2022
Problem
Sides 
, 
, 
, and 
, respectively of convex quadrilateral 
are extended past 
, 
, 
, and 
to points 
, 
, 
, and 
. If 
, 
, 
, and 
, and the area of is 10, determine the area of quadrilateral 
.
$$ (Error compiling LaTeX. Unknown error_msg) [asy] unitsize(1 cm);
pair[] A, B, C, D;
A[0] = (0,0); B[0] = (0.6,1.2); C[0] = (-0.3,2.5); D[0] = (-1.5,0.7); B[1] = interp(A[0],B[0],2); C[1] = interp(B[0],C[0],2); D[1] = interp(C[0],D[0],2); A[1] = interp(D[0],A[0],2);
draw(A[1]--B[1]--C[1]--D[1]--cycle); draw(A[0]--B[1]); draw(B[0]--C[1]); draw(C[0]--D[1]); draw(D[0]--A[1]);
label("", A[0], SW);
label("", B[0], SE);
label("", C[0], NE);
label("", D[0], NW);
label("", A[1], SE);
label("", B[1], NE);
label("", C[1], N);
label("", D[1], SW);
[\asy] $$ (Error compiling LaTeX. Unknown error_msg)
Solution
Notice that the area of 
is the same as that of 
(same base, same height). Thus, the area of is twice that (same height, twice the base). Similarly, [ 
] = 2 
[ 
], and so on.
Adding all of these, we see that the area the four triangles around is twice [ ] + [ ] + [ 
] + [ 
], which is itself twice the area of the quadrilateral . Finally, [] = [] + 4 [] = 5 [] = 
.
~ Mathavi
Note: Anyone with a diagram would be of great help (still new to LaTex).
