To find the ratio of the area of the smaller triangle to the area of the larger triangle, we need to calculate the areas of both triangles.
The formula for the area of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Larger Triangle
Vertices: \((5, 9)\), \((2, 3)\), \((5, 3)\)
Assigning:
- \((x_1, y_1) = (5, 9)\)
- \((x_2, y_2) = (2, 3)\)
- \((x_3, y_3) = (5, 3)\)
Calculating the area:
\[ \text{Area}_{\text{large}} = \frac{1}{2} \left| 5(3 - 3) + 2(3 - 9) + 5(9 - 3) \right| \]
Breaking it down:
\[ = \frac{1}{2} \left| 5(0) + 2(-6) + 5(6) \right| \] \[ = \frac{1}{2} \left| 0 - 12 + 30 \right| \] \[ = \frac{1}{2} \left| 18 \right| = 9 \]
Smaller Triangle
Vertices: \((12, 6)\), \((10, 2)\), \((12, 2)\)
Assigning:
- \((x_1, y_1) = (12, 6)\)
- \((x_2, y_2) = (10, 2)\)
- \((x_3, y_3) = (12, 2)\)
Calculating the area:
\[ \text{Area}_{\text{small}} = \frac{1}{2} \left| 12(2 - 2) + 10(2 - 6) + 12(6 - 2) \right| \]
Breaking it down:
\[ = \frac{1}{2} \left| 12(0) + 10(-4) + 12(4) \right| \] \[ = \frac{1}{2} \left| 0 - 40 + 48 \right| \] \[ = \frac{1}{2} \left| 8 \right| = 4 \]
Ratio of Areas
Now we find the ratio of the area of the smaller triangle to the area of the larger triangle:
\[ \text{Ratio} = \frac{\text{Area}{\text{small}}}{\text{Area}{\text{large}}} = \frac{4}{9} \]
To express this as a decimal:
\[ \text{Decimal Ratio} \approx 0.4444 \text{ (repeating)} \]
Thus, the ratio of the area of the smallest triangle in relation to the area of the largest triangle is approximately 0.4444.