Final answer:
The perimeter of triangle XYZ is 14.61 units and the area is approximately 8.09 square units, using the distance formula and Heron's formula.
Step-by-step explanation:
To find the perimeter and area of a triangle with vertices X (0,2), Y(4,-1), and Z (-2,-1), we use the distance formula to calculate the lengths of the sides and then use Heron's formula for area.
Step 1: Calculate the lengths of the sides
For side XY:
Distance = √((x2-x1)² + (y2-y1)²)
= √((4-0)² + (-1-2)²)
= √(4² + (-3)²)
= √(16 + 9)
= √25
= 5 units
For side YZ:
Distance = √((x3-x2)² + (y3-y2)²)
= √((-2-4)² + (-1+1)²)
= √((-6)² + 0²)
= √(36)
= 6 units
For side ZX:
Distance = √((x1-x3)² + (y1-y3)²)
= √((0+2)² + (2+1)²)
= √(2² + 3²)
= √(4 + 9)
= √13
≈ 3.61 units
Step 2: Calculate the perimeter
Perimeter (P) = XY + YZ + ZX
= 5 + 6 + 3.61
= 14.61 units
Step 3: Calculate the area using Heron's formula
Semi-perimeter (s) = P/2
= 14.61/2
= 7.305 units
Area (A) = √(s(s-XY)(s-YZ)(s-ZX))
= √(7.305(7.305-5)(7.305-6)(7.305-3.61))
= √(7.305(2.305)(1.305)(3.695))
≈ √(65.45)
≈ 8.09 square units