Final answer:
The perimeter and area of triangle PQR are found by calculating the side lengths using the distance formula, summing them for the perimeter, and using Heron's formula to find the area.
Step-by-step explanation:
To find the perimeter and area of triangle PQR with vertices P(-1, 3), Q(-3, -1), and R(4, -1), we first calculate the lengths of the sides using the distance formula:
- PQ = √[(-3 - (-1))^2 + (-1 - 3)^2] = √[4 + 16] = √20
- QR = √[(4 - (-3))^2 + (-1 - (-1))^2] = √[49 + 0] = 7
- RP = √[(4 - (-1))^2 + (-1 - 3)^2] = √[25 + 16] = √41
Once we have the side lengths, the perimeter (P) is the sum of all sides: P = PQ + QR + RP.
To find the area (A), we can use Heron's formula, which states that if s is the semi-perimeter of the triangle (P/2), then:
A = √[s(s - PQ)(s - QR)(s - RP)]