Question

what is the area of the ratio of the smaller triangle to the larger triangle
smaller: 12,2 - 12,6 - 10,2
larger: 5,9 - 5,3 - 2,3

Answer 1

To find the area of the ratio of the smaller triangle to the larger triangle given the vertices of both triangles, we first need to calculate the area of each triangle using the coordinates of their vertices.

### Area of a Triangle Using Coordinates:

The formula for the area \(A\) of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is:

\[
A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|
\]

### Step 1: Calculate the area of the smaller triangle with vertices \( (12,2), (12,6), (10,2) \)

Assigning the coordinates:
- \( (x_1, y_1) = (12, 2) \)
- \( (x_2, y_2) = (12, 6) \)
- \( (x_3, y_3) = (10, 2) \)

Substituting into the area formula:

\[
A = \frac{1}{2} \left| 12(6-2) + 12(2-2) + 10(2-6) \right|
\]
\[
= \frac{1}{2} \left| 12 \cdot 4 + 0 + 10 \cdot (-4) \right|
\]
\[
= \frac{1}{2} \left| 48 - 40 \right|
\]
\[
= \frac{1}{2} \left| 8 \right| = 4
\]

### Step 2: Calculate the area of the larger triangle with vertices \( (5,9), (5,3), (2,3) \)

Assigning the coordinates:
- \( (x_1, y_1) = (5, 9) \)
- \( (x_2, y_2) = (5, 3) \)
- \( (x_3, y_3) = (2, 3) \)

Substituting into the area formula:

\[
A = \frac{1}{2} \left| 5(3-3) + 5(3-9) + 2(9-3) \right|
\]
\[
= \frac{1}{2} \left| 0 + 5 \cdot (-6) + 2 \cdot 6 \right|
\]
\[
= \frac{1}{2} \left| -30 + 12 \right|
\]
\[
= \frac{1}{2} \left| -18 \right| = 9
\]

### Step 3: Calculate the ratio of the areas

The area of the smaller triangle is \(4\) and the area of the larger triangle is \(9\). The ratio of the area of the smaller triangle to the larger triangle is:

\[
\text{Ratio} = \frac{\text{Area}_{\text{smaller}}}{\text{Area}_{\text{larger}}} = \frac{4}{9}
\]

### Conclusion

Thus, the area of the ratio of the smaller triangle to the larger triangle is \(\frac{4}{9}\).