Alright, let's break down the problem step by step.
### Step 1: Define the Vertices of the Original Hexagon
We are given the vertices of the regular hexagon ABCDEF:
- A(4, 4√3)
- B(8, 4√3)
- C(10, 2√3)
- D(8, 0)
- E(4, 0)
- F(2, 2√3)
### Step 2: Calculate the Side Length of the Original Hexagon
To find the side length of the hexagon, we can use the distance formula between two adjacent vertices, say A and B:
[tex]\[ \text{side\_length\_original} = \sqrt{(8 - 4)^2 + (4\sqrt{3} - 4\sqrt{3})^2} = \sqrt{(4)^2 + (0)^2} = \sqrt{16} = 4 \][/tex]
### Step 3: Calculate the Perimeter of the Original Hexagon
Since a regular hexagon has all sides of equal length, the perimeter can be calculated by multiplying the side length by 6:
[tex]\[ \text{perimeter\_original} = 6 \times 4 = 24 \][/tex]
### Step 4: Calculate the Area of the Original Hexagon
The area of a regular hexagon can be calculated using the formula:
[tex]\[ \text{area\_original} = \frac{3\sqrt{3}}{2} \times (\text{side\_length\_original})^2 \][/tex]
Plugging in the side length:
[tex]\[ \text{area\_original} = \frac{3\sqrt{3}}{2} \times (4)^2 = \frac{3\sqrt{3}}{2} \times 16 = 24\sqrt{3}/2 = 41.569219381653056 \][/tex]
### Step 5: Determine the Side Length of the Smaller Hexagon
As the sides of the hexagon are reduced by 40%, the side length of the smaller hexagon is 60% of the original:
[tex]\[ \text{side\_length\_smaller} = 0.6 \times 4 = 2.4 \][/tex]
### Step 6: Calculate the Perimeter of the Smaller Hexagon
With the new side length, the perimeter of the smaller hexagon can be calculated similarly by multiplying the side length by 6:
[tex]\[ \text{perimeter\_smaller} = 6 \times 2.4 = 14.4 \][/tex]
### Step 7: Calculate the Area of the Smaller Hexagon
Using the area formula for a regular hexagon again, but with the new side length:
[tex]\[ \text{area\_smaller} = \frac{3\sqrt{3}}{2} \times (2.4)^2 = \frac{3\sqrt{3}}{2} \times 5.76 = 15.0 \][/tex]
### Final Result:
So, the perimeter and area of the smaller hexagon, rounded to the nearest tenth, are:
- Perimeter: 14.4
- Area: 15.0
These results are derived step-by-step considering the given transformations and properties of regular hexagons.
