We are given a triangle with vertices at the points $(0, 6.8)$, $(4.5, 6.8)$, and $(2.25, 0)$ on a coordinate plane. Each unit represents 1 centimeter. The goal is to find the approximate perimeter of the triangle to the nearest tenth of a centimeter.

Step 1: Compute the lengths of the sides of the triangle

We will use the distance formula to find the lengths of each side of the triangle. The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

Side 1: Between points $(0, 6.8)$ and $(4.5, 6.8)$

Since the y-coordinates are the same, the distance is simply the difference in the x-coordinates: $d_1 = |4.5 - 0| = 4.5 \, \text{cm}$

Side 2: Between points $(0, 6.8)$ and $(2.25, 0)$

$d_2 = \sqrt{(2.25 - 0)^2 + (0 - 6.8)^2} = \sqrt{2.25^2 + (-6.8)^2}$

Side 3: Between points $(4.5, 6.8)$ and $(2.25, 0)$

$d_3 = \sqrt{(2.25 - 4.5)^2 + (0 - 6.8)^2} = \sqrt{(-2.25)^2 + (-6.8)^2}$

Step 2: Calculate the perimeter

The perimeter $P$ of the triangle is the sum of the lengths of all three sides: $P = d_1 + d_2 + d_3$

I will calculate this now.The approximate perimeter of the triangle is 18.8 centimeters, rounded to the nearest tenth.

Would you like further clarification or details? Here are some related questions you might find interesting:

1. How do you calculate the area of this triangle using its vertices?
2. How does the distance formula relate to the Pythagorean theorem?
3. What are other methods to find the perimeter of a triangle given vertices in different units?
4. How does changing the coordinates of one vertex affect the perimeter?
5. Can this triangle be classified as isosceles or equilateral based on the side lengths?

Tip: The distance formula is a powerful tool in geometry for measuring lengths between points on a plane. It's derived directly from the Pythagorean theorem!