To solve for the perimeter and area of triangle △ABC given its vertices A(2, 8), B(16, 2), and C(6, 2), let's break the problem down step by step.
### Step 1: Calculate the lengths of the sides
We use the distance formula to calculate the distances between each pair of points. The distance formula between two points (x1, y1) and (x2, y2) is:
[tex]\[ \text{Distance} = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \][/tex]
#### Side AB
Given points A(2, 8) and B(16, 2):
[tex]\[ AB = \sqrt{(16 - 2)^2 + (2 - 8)^2} \][/tex]
[tex]\[ AB = \sqrt{(14)^2 + (-6)^2} \][/tex]
[tex]\[ AB = \sqrt{196 + 36} \][/tex]
[tex]\[ AB = \sqrt{232} \][/tex]
[tex]\[ AB \approx 15.23 \][/tex]
#### Side BC
Given points B(16, 2) and C(6, 2):
[tex]\[ BC = \sqrt{(6 - 16)^2 + (2 - 2)^2} \][/tex]
[tex]\[ BC = \sqrt{(-10)^2 + (0)^2} \][/tex]
[tex]\[ BC = \sqrt{100} \][/tex]
[tex]\[ BC = 10 \][/tex]
#### Side CA
Given points C(6, 2) and A(2, 8):
[tex]\[ CA = \sqrt{(2 - 6)^2 + (8 - 2)^2} \][/tex]
[tex]\[ CA = \sqrt{(-4)^2 + (6)^2} \][/tex]
[tex]\[ CA = \sqrt{16 + 36} \][/tex]
[tex]\[ CA = \sqrt{52} \][/tex]
[tex]\[ CA \approx 7.21 \][/tex]
### Step 2: Calculate the perimeter
The perimeter [tex]\( P \)[/tex] of the triangle is the sum of the lengths of its sides:
[tex]\[ P = AB + BC + CA \][/tex]
[tex]\[ P \approx 15.23 + 10 + 7.21 \][/tex]
[tex]\[ P \approx 32.44 \][/tex]
### Step 3: Calculate the area
We can use the formula for the area of a triangle given its vertices (using determinant method):
[tex]\[ \text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_1 - y_1x_2 - y_2x_3 - y_3x_1 \right| \][/tex]
Substitute in the points A(2, 8), B(16, 2), C(6, 2):
[tex]\[ \text{Area} = \frac{1}{2} \left| 2 \cdot 2 + 16 \cdot 2 + 6 \cdot 8 - 8 \cdot 16 - 2 \cdot 6 - 2 \cdot 2 \right| \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \left| 4 + 32 + 48 - 128 - 12 - 4 \right| \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \left| 84 - 144 \right| \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \left| -60 \right| \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \cdot 60 \][/tex]
[tex]\[ \text{Area} = 30 \][/tex]
### Final Answers
- The perimeter of △ABC is approximately [tex]\( 32.44 \)[/tex] units.
- The area of △ABC is [tex]\( 30 \)[/tex] square units.
These steps show how we arrive at the final answers for both the perimeter and area of the given triangle.
