To find the coordinates of point $DD$D that divides the line segment $ACAC$AC in the ratio $1:31:3$1:3:

Using the section formula, the coordinates of $DD$D are:

$D(\frac{1\cdot 6+3\cdot 0}{1+3},\frac{1\cdot 0+3\cdot 4}{1+3})=D(\frac{6}{4},\frac{12}{4})=D(1.5,3)D\backslash left(\; \backslash frac\{1\; \backslash cdot\; 6\; +\; 3\; \backslash cdot\; 0\}\{1\; +\; 3\},\; \backslash frac\{1\; \backslash cdot\; 0\; +\; 3\; \backslash cdot\; 4\}\{1\; +\; 3\}\; \backslash right)\; =\; D\backslash left(\; \backslash frac\{6\}\{4\},\; \backslash frac\{12\}\{4\}\; \backslash right)\; =\; D(1.5,\; 3)$D(1+31⋅6+3⋅0​,1+31⋅0+3⋅4​)=D(46​,412​)=D(1.5,3)

Next, we calculate the area of triangles $ABDABD$ABD and $ABCABC$ABC:

1. Area of triangle $ABCABC$ABC:

${ABC}_{}=\frac{1}{2}\mid 0(3-0)+1(0-4)+6(4-3)\mid =\frac{1}{2}\mid 0-4+6\mid =\frac{1}{2}\times 2=1\backslash text\{Area\}\_\{ABC\}\; =\; \backslash frac\{1\}\{2\}\; \backslash left|\; 0(3-0)\; +\; 1(0-4)\; +\; 6(4-3)\; \backslash right|\; =\; \backslash frac\{1\}\{2\}\; \backslash left|\; 0\; –\; 4\; +\; 6\; \backslash right|\; =\; \backslash frac\{1\}\{2\}\; \backslash times\; 2\; =\; 1$AreaABC​=21​∣0(3−0)+1(0−4)+6(4−3)∣=21​∣0−4+6∣=21​×2=1

1. Area of triangle $ABDABD$ABD:

${ABD}_{}=\frac{1}{2}\mid 0(3-3)+1(3-4)+1.5(4-3)\mid =\frac{1}{2}\mid 0-1+1.5\mid =\frac{1}{2}\times 0.5=0.25\backslash text\{Area\}\_\{ABD\}\; =\; \backslash frac\{1\}\{2\}\; \backslash left|\; 0(3-3)\; +\; 1(3-4)\; +\; 1.5(4-3)\; \backslash right|\; =\; \backslash frac\{1\}\{2\}\; \backslash left|\; 0\; –\; 1\; +\; 1.5\; \backslash right|\; =\; \backslash frac\{1\}\{2\}\; \backslash times\; 0.5\; =\; 0.25$AreaABD​=21​∣0(3−3)+1(3−4)+1.5(4−3)∣=21​∣0−1+1.5∣=21​×0.5=0.25

Thus, the ratio of the areas is:

${ABD}_{}:{ABC}_{}=0.25:1=1:4\backslash text\{Area\}\_\{ABD\}\; :\; \backslash text\{Area\}\_\{ABC\}\; =\; 0.25\; :\; 1\; =\; 1\; :\; 4$AreaABD​:AreaABC​=0.25:1=1:4

Conclusion: The coordinates of point $DD$D are $(1.5,3)(1.5,\; 3)$(1.5,3), and the area ratio of triangle $ABDABD$ABD to triangle $ABCABC$ABC is $1:41:4$1:4.