Medial triangle area from coordinates: For A(0, −1), B(0, 3), and C(2, 1), let △₁ be the area of triangle ABC and △₂ be the area of the triangle formed by the midpoints of its sides. Given the standard property △₂ = (1/4)△₁, find △₂ (correcting the inconsistent “△₁ = 1” note by computing from coordinates).

Introduction / Context:The medial triangle (joining midpoints of the sides) has area exactly one quarter of the original triangle’s area, regardless of the triangle’s shape. We will compute △₁ from coordinates and then take a quarter to get △₂.

Given Data / Assumptions:

• A(0, −1), B(0, 3), C(2, 1).
• Standard coordinate area formula applies.
• We ignore the inconsistent “△₁ = 1” note and compute △₁ correctly.

Concept / Approach:Area(ABC) = (1/2)*|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. Then △₂ = (1/4)△₁.

Step-by-Step Solution:

Verification / Alternative check:The midpoint triangle always has sides parallel to the original and each side half the length, making the area one fourth.

Why Other Options Are Wrong:2, 3, 4, 5 do not equal one quarter of 4.

Common Pitfalls:Accepting the inconsistent “△₁ = 1” without recomputation; arithmetic slips in the coordinate formula.

Final Answer:1.