The points given are $(0, 4)$, $(2, -2)$, and $(4, 2)$. We can calculate the area and perimeter of the triangle formed by these points.

Step 1: Calculate the Perimeter

To find the perimeter, we calculate the distance between each pair of points using the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Distance between $(0, 4)$ and $(2, -2)$:

$d_1 = \sqrt{(2 - 0)^2 + (-2 - 4)^2} = \sqrt{2^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} = 2\sqrt{10}$

• Distance between $(2, -2)$ and $(4, 2)$:

$d_2 = \sqrt{(4 - 2)^2 + (2 + 2)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$

• Distance between $(4, 2)$ and $(0, 4)$:

$d_3 = \sqrt{(4 - 0)^2 + (2 - 4)^2} = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}$

Thus, the perimeter $P$ is:

$P = d_1 + d_2 + d_3 = 2\sqrt{10} + 2\sqrt{5} + 2\sqrt{5} = 2\sqrt{10} + 4\sqrt{5}$

Step 2: Calculate the Area

The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ can be calculated using the formula:

$\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$

Substitute the coordinates of the points $(0, 4)$, $(2, -2)$, and $(4, 2)$:

$\text{Area} = \frac{1}{2} \left| 0(-2 - 2) + 2(2 - 4) + 4(4 + 2) \right|$

$= \frac{1}{2} \left| 0 + 2(-2) + 4(6) \right| = \frac{1}{2} \left| -4 + 24 \right| = \frac{1}{2} \times 20 = 10$

Final Results:

• Perimeter: $2\sqrt{10} + 4\sqrt{5}$
• Area: 10 square units

Would you like more details or have any questions?

Here are some related questions:

1. How do we calculate the perimeter of more complex polygons?
2. What is the significance of the area formula for triangles in geometry?
3. Can the distance formula be applied to three-dimensional geometry?
4. How does the method for finding the area differ for a non-triangular shape?
5. How do changes in the coordinates affect the perimeter and area?

Tip: Always double-check calculations for distance when working with the Pythagorean theorem!