Let A(0, -1), B(0, 3) and C(2, 1) be three points. Let Δ1 be the area of triangle ABC and Δ2 be the area of the triangle formed by joining the midpoints of sides AB, BC and CA. If the ratio Δ1 : Δ2 = x : 1, find the value of x.

Introduction / Context:
This problem explores a standard result in coordinate geometry and triangle geometry. When you join the midpoints of the sides of a triangle, you obtain a new triangle called the medial triangle. There is a fixed relationship between the area of the original triangle and the area of the medial triangle, which you can derive from either coordinate geometry or basic geometry of similar triangles.

Given Data / Assumptions:

• Vertices of triangle ABC are A(0, -1), B(0, 3), and C(2, 1)
• Δ1 is area of triangle ABC
• Δ2 is area of triangle formed by midpoints of AB, BC, and CA
• We are told that Δ1 : Δ2 = x : 1

Concept / Approach:
Joining the midpoints of sides of any triangle produces a triangle similar to the original triangle with a similarity ratio of 1 : 2 (the medial triangle has half the side length of the original). Since area scales as the square of the similarity ratio, the area of the medial triangle is 1/4 of the area of the original triangle. Therefore, the ratio Δ1 : Δ2 is 4 : 1. We do not even need to compute exact coordinates of midpoints or actual numeric areas to get the ratio.

Step-by-Step Solution:
Step 1: Recognise that the triangle formed by joining midpoints of sides is the medial triangle. Step 2: Each side of the medial triangle is parallel to a side of the original triangle and exactly half its length. Step 3: Similarity ratio of side lengths (medial : original) = 1 : 2. Step 4: Area ratio is the square of the side ratio: (1/2)^2 = 1/4. Step 5: Therefore Δ2 = (1/4) * Δ1. Step 6: Rearranging, Δ1 : Δ2 = 4 : 1, so x = 4.

Verification / Alternative check:
You can verify by explicitly computing the coordinates of the midpoints and using the coordinate geometry area formula. However, this will still yield the same 1/4 area factor. Since the result holds for any triangle, the exact coordinates do not affect the ratio, only confirm it numerically.

Why Other Options Are Wrong:
2 and 3: These would suggest the medial triangle has half or one third of the area, which contradicts the similarity rule. 5 and 6: These give even larger ratios and are not consistent with the side length halving.

Common Pitfalls:
Assuming that halving the sides halves the area instead of reducing it to one quarter. Confusing the mid-segment triangle with some other internal triangle like the centroid joined triangle. Trying long coordinate calculations instead of using the known geometric result.

Final Answer:
The value of x is 4