27. $$ D(0,2), E(-1,5), G(4,0) $$

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It looks like you have three points in the Cartesian plane:

- **D** $$(0, 2)$$
- **E** $$(-1, 5)$$
- **G** $$(4, 0)$$

Assuming you want to analyze the triangle formed by these three points, here's a comprehensive breakdown:

### 1. **Plotting the Points**

First, let's visualize the points and the triangle they form:

- **D(0, 2)**: Located on the y-axis, 2 units above the origin.
- **E(-1, 5)**: Located in the second quadrant, 1 unit left of the y-axis and 5 units above the origin.
- **G(4, 0)**: Located on the x-axis, 4 units to the right of the origin.

### 2. **Calculating the Side Lengths**

Using the distance formula between two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$:
$$
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

- **Distance DE** between D and E:
  $$
  \sqrt{(-1 - 0)^2 + (5 - 2)^2} = \sqrt{1 + 9} = \sqrt{10} \approx 3.16 \text{ units}
  $$

- **Distance EG** between E and G:
  $$
  \sqrt{(4 - (-1))^2 + (0 - 5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \approx 7.07 \text{ units}
  $$

- **Distance GD** between G and D:
  $$
  \sqrt{(0 - 4)^2 + (2 - 0)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \approx 4.47 \text{ units}
  $$

### 3. **Calculating the Perimeter**

The perimeter $$ P $$ of triangle $$ \triangle DEG $$ is the sum of its side lengths:
$$
P = DE + EG + GD = \sqrt{10} + 5\sqrt{2} + 2\sqrt{5} \approx 3.16 + 7.07 + 4.47 = 14.70 \text{ units}
$$

### 4. **Calculating the Area**

The area of a triangle given three vertices can be found using the **shoelace formula**:
$$
\text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|
$$

Plugging in the coordinates:

$$
\begin{align*}
\text{Area} &= \frac{1}{2} |0(5 - 0) + (-1)(0 - 2) + 4(2 - 5)| \
&= \frac{1}{2} |0 + 2 - 12| \
&= \frac{1}{2} | -10 | \
&= \frac{1}{2} \times 10 \
&= 5 \text{ square units}
\end{align*}
$$

### 5. **Centroid of the Triangle**

The centroid $$ (C_x, C_y) $$ is the average of the vertices' coordinates:
$$
C_x = \frac{0 + (-1) + 4}{3} = \frac{3}{3} = 1
$$
$$
C_y = \frac{2 + 5 + 0}{3} = \frac{7}{3} \approx 2.33
$$
So, the centroid is at $$ (1, \frac{7}{3}) $$.

### 6. **Slope of Each Side**

Calculating the slope $$ m $$ between two points:
$$
m = \frac{y_2 - y_1}{x_2 - x_1}
$$

- **Slope DE**:
  $$
  m_{DE} = \frac{5 - 2}{-1 - 0} = \frac{3}{-1} = -3
  $$

- **Slope EG**:
  $$
  m_{EG} = \frac{0 - 5}{4 - (-1)} = \frac{-5}{5} = -1
  $$

- **Slope GD**:
  $$
  m_{GD} = \frac{2 - 0}{0 - 4} = \frac{2}{-4} = -0.5
  $$

### 7. **Checking for Collinearity**

If all three points lie on a straight line, the slopes between each pair should be equal. Here:

$$
m_{DE} = -3 \neq m_{EG} = -1 \neq m_{GD} = -0.5
$$

Thus, the points **D**, **E**, and **G** are **not collinear** and indeed form a **triangle**.

### **Summary**

- **Side Lengths**:
  - DE: $$ \sqrt{10} \approx 3.16 $$ units
  - EG: $$ 5\sqrt{2} \approx 7.07 $$ units
  - GD: $$ 2\sqrt{5} \approx 4.47 $$ units

- **Perimeter**: Approximately $$ 14.70 $$ units

- **Area**: $$ 5 $$ square units

- **Centroid**: $$ \left(1, \frac{7}{3}\right) $$

- **Slopes**:
  - DE: $$ -3 $$
  - EG: $$ -1 $$
  - GD: $$ -0.5 $$

If you have a more specific question regarding these points or require further analysis, feel free to ask!
The triangle formed by points D(0,2), E(-1,5), and G(4,0) has side lengths approximately 3.16 units, 7.07 units, and 4.47 units. The perimeter is about 14.70 units, and the area is 5 square units. The centroid is at (1, 7/3). The slopes of the sides are -3, -1, and -0.5, respectively.

27. \( D(0,2), E(-1,5), G(4,0) \)

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