To find the coordinates of vertex A, we start with the formula for the area of a triangle given vertices $A(x_1,y_1)A(x\_1,\; y\_1)$A(x1​,y1​), $B(1,-2)B(1,\; -2)$B(1,−2), and $C(2,3)C(2,\; 3)$C(2,3):

$Area=\frac{1}{2}\mathrm{\mid }{x}_1(-2-3)+1(3-y_1)+2(y_1+2)\mathrm{\mid }=8\backslash text\{Area\}\; =\; \backslash frac\{1\}\{2\}\; |\; x\_1(-2\; –\; 3)\; +\; 1(3\; –\; y\_1)\; +\; 2(y\_1\; +\; 2)\; |\; =\; 8$Area=21​∣x1​(−2−3)+1(3−y1​)+2(y1​+2)∣=8Simplifying gives:

$\mathrm{\mid }-5x_1+3-y_1+2y_1+4\mathrm{\mid }=16|\; -5x\_1\; +\; 3\; –\; y\_1\; +\; 2y\_1\; +\; 4\; |\; =\; 16$∣−5x1​+3−y1​+2y1​+4∣=16 $\mathrm{\mid }-5x_1+y_1+7\mathrm{\mid }=16|\; -5x\_1\; +\; y\_1\; +\; 7\; |\; =\; 16$∣−5x1​+y1​+7∣=16This results in two equations:

1. $-5x_1+y_1+7=16-5x\_1\; +\; y\_1\; +\; 7\; =\; 16$−5x1​+y1​+7=16
2. $-5x_1+y_1+7=-16-5x\_1\; +\; y\_1\; +\; 7\; =\; -16$−5x1​+y1​+7=−16

From the first equation:

$y_1=5x_1+9y\_1\; =\; 5x\_1\; +\; 9$y1​=5x1​+9From the second equation:

$y_1=5x_1-23y\_1\; =\; 5x\_1\; –\; 23$y1​=5x1​−23Next, we substitute these into the line equation $2x+y-20=02x\; +\; y\; –\; 20\; =\; 0$2x+y−20=0:

1. For $y_1=5x_1+9y\_1\; =\; 5x\_1\; +\; 9$y1​=5x1​+9:

$2x_1+(5x_1+9)-20=0\hspace{0.25em} ⟹\hspace{0.25em} 7x_1-11=0\hspace{0.25em} ⟹\hspace{0.25em} x_1=\frac{11}{7},y_1=\frac{5\times 11}{7}+9=\frac{55}{7}+\frac{63}{7}=\frac{118}{7}2x\_1\; +\; (5x\_1\; +\; 9)\; –\; 20\; =\; 0\; \backslash implies\; 7x\_1\; –\; 11\; =\; 0\; \backslash implies\; x\_1\; =\; \backslash frac\{11\}\{7\},\; \backslash quad\; y\_1\; =\; \backslash frac\{5\; \backslash times\; 11\}\{7\}\; +\; 9\; =\; \backslash frac\{55\}\{7\}\; +\; \backslash frac\{63\}\{7\}\; =\; \backslash frac\{118\}\{7\}$2x1​+(5x1​+9)−20=0⟹7x1​−11=0⟹x1​=711​,y1​=75×11​+9=755​+763​=7118​

1. For $y_1=5x_1-23y\_1\; =\; 5x\_1\; –\; 23$y1​=5x1​−23:

$2x_1+(5x_1-23)-20=0\hspace{0.25em} ⟹\hspace{0.25em} 7x_1-43=0\hspace{0.25em} ⟹\hspace{0.25em} x_1=\frac{43}{7},y_1=\frac{5\times 43}{7}-23=\frac{215}{7}-\frac{161}{7}=\frac{54}{7}2x\_1\; +\; (5x\_1\; –\; 23)\; –\; 20\; =\; 0\; \backslash implies\; 7x\_1\; –\; 43\; =\; 0\; \backslash implies\; x\_1\; =\; \backslash frac\{43\}\{7\},\; \backslash quad\; y\_1\; =\; \backslash frac\{5\; \backslash times\; 43\}\{7\}\; –\; 23\; =\; \backslash frac\{215\}\{7\}\; –\; \backslash frac\{161\}\{7\}\; =\; \backslash frac\{54\}\{7\}$2x1​+(5x1​−23)−20=0⟹7x1​−43=0⟹x1​=743​,y1​=75×43​−23=7215​−7161​=754​Thus, the coordinates of vertex A are either $(\frac{11}{7},\frac{118}{7})\backslash left(\; \backslash frac\{11\}\{7\},\; \backslash frac\{118\}\{7\}\; \backslash right)$(711​,7118​) or $(\frac{43}{7},\frac{54}{7})\backslash left(\; \backslash frac\{43\}\{7\},\; \backslash frac\{54\}\{7\}\; \backslash right)$(743​,754​).