Triangles ABC and ADE have areas 2007 and 7002 respectively, with B=(0,0), C=(223,0), D=(680,380), and E=(689,389). What is the sum of all possible x-coordinates of A?
+26367 Triangles ABC and ADE have areas \(2007 \)and \(7002 \)respectively, with
\(B=(0,0)\),
\(C=(223,0)\),
\(D=(680,380)\), and
\(E=(689,389)\).
What is the sum of all possible x-coordinates of A?
My attempt:
I use the Gauss's shoelace area formula.
\(\text{First solution for $y_A$} :\\ \begin{array}{rcl} \mathbf{\text{2[ABC]} } = \begin{array}{|rr|} x_A & y_A \\ 0 & 0 \\ 223 & 0 & \\ x_A & y_A \\ \end{array} &=& 2* 2007 \\\\ \begin{array}{|rr|} x_A & y_A \\ 0 & 0 \\ 223 & 0 & \\ x_A & y_A \\ \end{array} &=& 2* 2007 \\ \begin{array}{|rr|} x_A*0-0*y_A + 0*0-223*0+223*y_A-x_A*0 \end{array} &=& 2* 2007 \\ 223*y_A&=& 2* 2007 \quad | \quad : 223 \\ \mathbf{y_A} &=& \mathbf{18} \\ \end{array} \\ \text{First solution for $x_A$}: \\ \begin{array}{rcl} \mathbf{\text{2[ADE]} } = \begin{array}{|rr|} x_A & 18 \\ 680 & 380 \\ 689 & 389 & \\ x_A & 18 \\ \end{array} &=& 2* 7002 \\\\ \begin{array}{|rr|} x_A & 18 \\ 680 & 380 \\ 689 & 389 & \\ x_A & 18 \\ \end{array} &=& 2* 7002 \\ \begin{array}{|rr|} 380x_A-680*18+680*389-689*380+689*18-389x_A \end{array} &=& 2* 2007 \\ -9x_A+680*371-689*362 &=& 2* 7002 \\ -9x_A+2862 &=& 2* 7002 \quad | \quad : (-9) \\ x_A - 318 &=& -1556\\ \mathbf{x_A} &=& \mathbf{-1238} \\ \end{array} \\ \text{Second solution for $x_A$}: \\ \begin{array}{rcl} \mathbf{\text{2[ADE]} } = \begin{array}{|rr|} x_A & 18 \\ 689 & 389 \\ 680 & 380 \\ x_A & 18 \\ \end{array} &=& 2* 7002 \\\\ \begin{array}{|rr|} x_A & 18 \\ 689 & 389 \\ 680 & 380 \\ x_A & 18 \\ \end{array} &=& 2* 7002 \\ \begin{array}{|rr|} 389x_A-689*18+689*380-680*389+680*18-380x_A \end{array} &=& 2* 2007 \\ 9x_A+689*362-689*371 &=& 2* 7002 \\ 9x_A-2862 &=& 2* 7002 \quad | \quad : 9 \\ x_A - 318 &=& 1556\\ \mathbf{x_A} &=& \mathbf{1874} \\ \end{array} \\ \text{Second solution for $y_A$}: \\ \begin{array}{rcl} \mathbf{\text{2[ABC]} } = \begin{array}{|rr|} x_A & y_A \\ 223 & 0 & \\ 0 & 0 \\ x_A & y_A \\ \end{array} &=& 2* 2007 \\\\ \begin{array}{|rr|} x_A & y_A \\ 223 & 0 & \\ 0 & 0 \\ x_A & y_A \\ \end{array} &=& 2* 2007 \\ \begin{array}{|rr|} x_A*0-223*y_A +223*0 - 0*0 + 0*y_A -x_A*0 \end{array} &=& 2* 2007 \\ -223*y_A&=& 2* 2007 \quad | \quad : (-223) \\ \mathbf{y_A} &=& \mathbf{-18} \\ \end{array} \\ \text{Third Solution for $x_A$}: \\ \begin{array}{rcl} \mathbf{\text{2[ADE]} } = \begin{array}{|rr|} x_A & -18 \\ 680 & 380 \\ 689 & 389 \\ x_A & -18 \\ \end{array} &=& 2* 7002 \\\\ \begin{array}{|rr|} x_A & -18 \\ 680 & 380 \\ 689 & 389 \\ x_A & -18 \\ \end{array} &=& 2* 7002 \\ \begin{array}{|rr|} 380x_A+680*18+680*389-689*380-689*18-389x_A \end{array} &=& 2* 2007 \\ -9x_A+680*407-689*398 &=& 2* 7002 \\ -9x_A+2538 &=& 2* 7002 \quad | \quad : (-9) \\ x_A - 282 &=& -1556\\ \mathbf{x_A} &=& \mathbf{-1274} \\ \end{array} \\ \text{Fourth Solution for $x_A$}: \\ \begin{array}{rcl} \mathbf{\text{2[ADE]} } = \begin{array}{|rr|} x_A & -18 \\ 689 & 389 \\ 680 & 380 \\ x_A & -18 \\ \end{array} &=& 2* 7002 \\\\ \begin{array}{|rr|} x_A & -18 \\ 689 & 389 \\ 680 & 380 \\ x_A & -18 \\ \end{array} &=& 2* 7002 \\ \begin{array}{|rr|} 389x_A+689*18+689*380-680*389-680*18-380x_A \end{array} &=& 2* 2007 \\ 9x_A+689*398-680*407 &=& 2* 7002 \\ 9x_A-2538 &=& 2* 7002 \quad | \quad : 9 \\ x_A - 282 &=& 1556\\ \mathbf{x_A} &=& \mathbf{1838} \\ \end{array} \\ \text{The sum of all possible x-coordinates of A is $-1238+1874-1274+1838 = \mathbf{1200}$} \)
