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

Answer 1

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

### Area of the larger triangle
The vertices of the larger triangle are:
- \( A(5, 9) \)
- \( B(2, 3) \)
- \( C(5, 3) \)

We can use the formula for the area of a triangle given 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 of the larger triangle:

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

### Area of the smaller triangle
The vertices of the smaller triangle are:
- \( D(12, 6) \)
- \( E(10, 2) \)
- \( F(12, 2) \)

Using the same area formula:

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

### Ratio of the areas
Now to 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 \( \frac{4}{9} \).