To find the ratio of the area of the smaller triangle to the area of the larger triangle, we first need to calculate the area of each triangle.
Area of the Larger Triangle
The vertices of the larger triangle are \((5,3)\), \((5,9)\), and \((2,3)\).
Using the formula for the area of a triangle given by vertices \((x_1, y_1)\), \((x_2, y_2)\), \((x_3, y_3)\):
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates into the formula:
\[ \text{Area}_{\text{larger}} = \frac{1}{2} \left| 5(9 - 3) + 5(3 - 3) + 2(3 - 9) \right| \]
Calculating inside the absolute value:
\[ = \frac{1}{2} \left| 5(6) + 5(0) + 2(-6) \right| \] \[ = \frac{1}{2} \left| 30 + 0 - 12 \right| \] \[ = \frac{1}{2} \left| 18 \right| = 9 \]
So, the area of the larger triangle is \(9\).
Area of the Smaller Triangle
The vertices of the smaller triangle are \((12,2)\), \((12,6)\), and \((10,2)\).
Using the same formula for the area:
\[ \text{Area}_{\text{smaller}} = \frac{1}{2} \left| 12(6 - 2) + 12(2 - 2) + 10(2 - 6) \right| \]
Calculating inside the absolute value:
\[ = \frac{1}{2} \left| 12(4) + 12(0) + 10(-4) \right| \] \[ = \frac{1}{2} \left| 48 + 0 - 40 \right| \] \[ = \frac{1}{2} \left| 8 \right| = 4 \]
So, the area of the smaller triangle is \(4\).
Ratio of Areas
Now we can find the ratio of the area of the smaller triangle to the area of the larger triangle:
\[ \text{Ratio} = \frac{\text{Area}{\text{smaller}}}{\text{Area}{\text{larger}}} = \frac{4}{9} \]
Thus, the ratio of the area of the smaller triangle to the area of the larger triangle is \(\frac{4}{9}\).