Section Formula

2:3, 1:1, 4:7, 3:2

For internal division, both parts of the ratio must be positive. Ratios like 5:0 or 0:3 are invalid because they would place the point at one of the endpoints, not between them.

((a+c)/2, (b+d)/2)

The midpoint formula states that for points (x₁, y₁) and (x₂, y₂), the midpoint is ((x₁+x₂)/2, (y₁+y₂)/2). Here x₁=a, y₁=b, x₂=c, y₂=d.

(4, 5)

Let third vertex be (x, y). Using centroid formula: 4 = (2+6+x)/3 → 12 = 8+x → x = 4. Similarly, 5 = (3+7+y)/3 → 15 = 10+y → y = 5. Therefore third vertex = (4, 5).

4

Using section formula for x-coordinate: x = (m×x₂ + n×x₁)/(m+n) = (3×8 + 4×1)/(3+4) = (24+4)/7 = 28/7 = 4.