1978 AHSME Problems/Problem 29
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).