The midpoint of a straight fence AB is M(−2, 5). One end of the fence is B(4, −1). Find the other end A.
Class 10Coordinate GeometryGeometry — Coordinates
A(-8, 11)
B(1, 2)
C(-8, 9)
D(8, -11)
Step-by-Step Solution
If M is the midpoint of AB then A=2M−B.x:2(−2)−4=−8; y:2(5)−(−1)=10+1=11. So A = (−8, 11), and the check works: (2−8+4,211+(−1))=(−2,5) ✓. Averaging M and B instead of doubling M gives (1, 2) — a point between them, not the far end. Writing 2(5)−1=9 loses the minus sign on B's y-coordinate.
Answer: A) (-8, 11)
Key idea: Section formula and midpoint
**Section formula (internal)**: A point P dividing segment AB in ratio m:n(P closer to A means m smaller) has coordinates:
P=(m+nmx2+nx1,m+nmy2+ny1)
Where A=(x₁, y₁), B=(x₂, y₂).
**Midpoint** is the special case m=n:M=(2x1+x2,2y1+y2).
**Example**: Find the point dividing AB in ratio 2:3 where A=(1,5),B=(6,−5).x=52⋅6+3⋅1=515=3.y=52⋅(−5)+3⋅5=55=1. Point: (3, 1).
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