Question

The larger triangle
Top (5,9) right point (2,3) left point (5,3)
The smaller triangle
Top (12,6) right point (10,2) left point (12,2)

What is the ratio of the area of the smallest triangle in relation to the area of the largest triangle as a decimal

Answer 1

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**.