Final answer:
After calculating the lengths of the sides of triangle PQR, it does not fit the criteria of the options provided. The triangle is neither a right triangle nor an isosceles triangle, as none of the sides follow the proportionalities of either PQ = QR or based on the Pythagorean theorem.
Step-by-step explanation:
To determine the type of triangle PQR with vertices P(-2,8), Q(2, 4), and R(4,6), we can calculate the lengths of the sides using the distance formula: distance = √((x2-x1)² + (y2-y1)²). Let's calculate the lengths of sides PQ, PR, and QR:
- PQ = √((2 - (-2))² + (4 - 8)²) = √(16 + 16) = √32 = 4√2
- PR = √((4 - (-2))² + (6 - 8)²) = √(36 + 4) = √40 = 2√10
- QR = √((4 - 2)² + (6 - 4)²) = √(4 + 4) = √8 = 2√2
Comparing the side lengths, we see that PQ (≈ 5.66) is less than PR (≈ 6.32) but is equal to QR (≈ 5.66). Therefore, triangle PQR is not a right triangle because the square of the length of one side does not equal the sum of the squares of the lengths of the other two sides, based on the Pythagorean theorem: a² + b² ≠ c². Additionally, triangle PQR cannot be an isosceles triangle with PQ = QR, and neither PQ > PR nor PQ < PR implies it is a right triangle.
Given this information, none of the provided options A, B, C, or D accurately describe triangle PQR.