To solve the problem step by step, we will follow the instructions given in the question:
### Step 1: Find the area of triangle ABC
The vertices of triangle ABC are given as:
- A(2, 2)
- B(4, 4)
- C(2, 6)
We can use the formula for the area of a triangle given its vertices (x1, y1), (x2, y2), (x3, y3):
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Substituting the coordinates of points A, B, and C into the formula:
\[
\text{Area} = \frac{1}{2} \left| 2(4 - 6) + 4(6 - 2) + 2(2 - 4) \right|
\]
Calculating each term:
\[
= \frac{1}{2} \left| 2(-2) + 4(4) + 2(-2) \right|
\]
\[
= \frac{1}{2} \left| -4 + 16 - 4 \right|
\]
\[
= \frac{1}{2} \left| 8 \right| = 4 \text{ square units}
\]
### Step 2: Find the midpoints D, E, and F
- **Midpoint D of AB**:
\[
D = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{2 + 4}{2}, \frac{2 + 4}{2} \right) = (3, 3)
\]
- **Midpoint E of BC**:
\[
E = \left( \frac{x_2 + x_3}{2}, \frac{y_2 + y_3}{2} \right) = \left( \frac{4 + 2}{2}, \frac{4 + 6}{2} \right) = (3, 5)
\]
- **Midpoint F of AC**:
\[
F = \left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2} \right) = \left( \frac{2 + 2}{2}, \frac{2 + 6}{2} \right) = (2, 4)
\]
### Step 3: Find the area of triangle DEF
Now we have the midpoints:
- D(3, 3)
- E(3, 5)
- F(2, 4)
Using the area formula again for triangle DEF:
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Substituting the coordinates of points D, E, and F:
\[
\text{Area} = \frac{1}{2} \left| 3(5 - 4) + 3(4 - 3) + 2(3 - 5) \right|
\]
Calculating each term:
\[
= \frac{1}{2} \left| 3(1) + 3(1) + 2(-2) \right|
\]
\[
= \frac{1}{2} \left| 3 + 3 - 4 \right|
\]
\[
= \frac{1}{2} \left| 2 \right| = 1 \text{ square unit}
\]
### Step 4: Find the ratio of the areas of triangle DEF to triangle ABC
Now we can find the ratio of the area of triangle DEF to the area of triangle ABC:
\[
\text{Ratio} = \frac{\text{Area of DEF}}{\text{Area of ABC}} = \frac{1}{4}
\]
Thus, the ratio is:
\[
1 : 4
\]
### Summary of the Solutions
(i) Area of triangle ABC = 4 square units
(ii) Area of triangle DEF = 1 square unit
(iii) Ratio of area of triangle DEF to area of triangle ABC = 1:4
