To find the area of triangle ABC with vertices A(0, -1), B(2, 1), and C(0, 3), we can use the formula for the area of a triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
### Step 1: Identify the coordinates
Let:
- \(A(0, -1)\) → \(x_1 = 0\), \(y_1 = -1\)
- \(B(2, 1)\) → \(x_2 = 2\), \(y_2 = 1\)
- \(C(0, 3)\) → \(x_3 = 0\), \(y_3 = 3\)
### Step 2: Substitute the coordinates into the area formula
Substituting the coordinates into the formula:
\[
\text{Area} = \frac{1}{2} \left| 0(1 - 3) + 2(3 - (-1)) + 0(-1 - 1) \right|
\]
### Step 3: Simplify the expression
Calculating each term:
- The first term: \(0(1 - 3) = 0\)
- The second term: \(2(3 + 1) = 2 \times 4 = 8\)
- The third term: \(0(-1 - 1) = 0\)
Putting it all together:
\[
\text{Area} = \frac{1}{2} \left| 0 + 8 + 0 \right| = \frac{1}{2} \times 8 = 4
\]
### Conclusion for Triangle ABC
Thus, the area of triangle ABC is \(4\) square units.
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### Step 4: Find the area of triangle formed by midpoints
Next, we need to find the area of the triangle formed by joining the midpoints of sides AB, BC, and AC.
#### Step 4.1: Calculate the midpoints
- Midpoint D of AB:
\[
D = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{0 + 2}{2}, \frac{-1 + 1}{2} \right) = (1, 0)
\]
- Midpoint E of BC:
\[
E = \left( \frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2} \right) = \left( \frac{2 + 0}{2}, \frac{1 + 3}{2} \right) = (1, 2)
\]
- Midpoint F of AC:
\[
F = \left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2} \right) = \left( \frac{0 + 0}{2}, \frac{-1 + 3}{2} \right) = (0, 1)
\]
#### Step 4.2: Calculate the area of triangle DEF
Using the same area formula for triangle DEF with vertices D(1, 0), E(1, 2), and F(0, 1):
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Substituting the coordinates:
\[
\text{Area} = \frac{1}{2} \left| 1(2 - 1) + 1(1 - 0) + 0(0 - 2) \right|
\]
Calculating each term:
- The first term: \(1(2 - 1) = 1\)
- The second term: \(1(1 - 0) = 1\)
- The third term: \(0(0 - 2) = 0\)
Putting it all together:
\[
\text{Area} = \frac{1}{2} \left| 1 + 1 + 0 \right| = \frac{1}{2} \times 2 = 1
\]
### Conclusion for Triangle DEF
Thus, the area of triangle DEF is \(1\) square unit.
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### Step 5: Show the ratio of the areas
Now, we can find the ratio of the areas of triangle ABC to triangle DEF:
\[
\text{Ratio} = \frac{\text{Area of } ABC}{\text{Area of } DEF} = \frac{4}{1} = 4:1
\]
### Final Conclusion
The ratio of the areas of triangle ABC to triangle DEF is \(4:1\).
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