Question

A big triangle with vertices (5,3) , (5,9) and (2,3)

A smaller triangle with vertices (12,2) , (12,6) and (10,2)


What is the Tati of the area of the smaller triangle in relation to the area of the larger 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 area of each triangle.

### Area of the Larger Triangle

The vertices of the larger triangle are \((5,3)\), \((5,9)\), and \((2,3)\).

Using the formula for the area of a triangle given by 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 into the formula:

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

Calculating inside the absolute value:

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

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

### Area of the Smaller Triangle

The vertices of the smaller triangle are \((12,2)\), \((12,6)\), and \((10,2)\).

Using the same formula for the area:

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

Calculating inside the absolute value:

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

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

### Ratio of Areas

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

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

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