Question

What is the ratio of the area of the smaller triangle, (10,2), (12,6), (12,2) in relation to the area of the larger triangle (2,3), (5,9), (5,3)

Answer 1

To find the ratio of the areas of the two triangles, we'll first compute the area of each triangle using the formula for the area of a triangle defined by its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

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

### Area of the Smaller Triangle (10,2), (12,6), (12,2)

Let’s denote the vertices as follows:
- \((x_1, y_1) = (10, 2)\)
- \((x_2, y_2) = (12, 6)\)
- \((x_3, y_3) = (12, 2)\)

Calculating the area:

\[
\text{Area} = \frac{1}{2} \left| 10(6 - 2) + 12(2 - 2) + 12(2 - 6) \right|
\]

Now, simplify:

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

So, the area of the smaller triangle is \(4\).

### Area of the Larger Triangle (2,3), (5,9), (5,3)

Let’s denote the vertices as follows:
- \((x_1, y_1) = (2, 3)\)
- \((x_2, y_2) = (5, 9)\)
- \((x_3, y_3) = (5, 3)\)

Calculating the area:

\[
\text{Area} = \frac{1}{2} \left| 2(9 - 3) + 5(3 - 3) + 5(3 - 9) \right|
\]

Now, simplify:

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

So, the area of the larger triangle is \(9\).

### Finding the Ratio of the Areas

Now we can find the ratio of the area of the smaller triangle to the area of the larger triangle:

\[
\text{Ratio} = \frac{\text{Area of Smaller Triangle}}{\text{Area of Larger Triangle}} = \frac{4}{9}
\]

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