Find the midpoint of segment joining (3,4) and (7,8).
- (a)(3, 4)
- (b)(5, 6)
- (c)(4, 5)
- (d)(6, 7)
Answer
Why
Correct — B. The midpoint averages the coordinates: ((x₁ + x₂)⁄2, (y₁ + y₂)⁄2).
x-coordinate: (3 + 7) ÷ 2 = 10 ÷ 2 = 5
y-coordinate: (4 + 8) ÷ 2 = 12 ÷ 2 = 6
Midpoint = (5, 6) → option (b)
Why the others are wrong
- (a)(3, 4) — (3, 4) is one of the two endpoints, not the point halfway between them.
- (c)(4, 5) — (4, 5) is only a quarter of the way from (3, 4): each coordinate has moved 1 of its 4 units. It divides the segment 1 : 3, not 1 : 1.
- (d)(6, 7) — (6, 7) is three quarters of the way from (3, 4), dividing the segment 3 : 1. Its distances to the two ends are unequal, so it is not the midpoint.
Concept
The midpoint formula is the section formula with ratio 1 : 1: average the x-coordinates, then average the y-coordinates.
A point dividing the join of (x₁, y₁) and (x₂, y₂) internally in the ratio m : n is ((mx₂ + nx₁)⁄(m + n), (my₂ + ny₁)⁄(m + n)). Put m = n and it becomes the plain average.
Key facts
- Midpoint of (x₁, y₁) and (x₂, y₂) = ((x₁ + x₂)⁄2, (y₁ + y₂)⁄2).
- Internal division in the ratio m : n gives ((mx₂ + nx₁)⁄(m + n), (my₂ + ny₁)⁄(m + n)).
- The centroid of a triangle divides each median in the ratio 2 : 1 from the vertex.
Study next
Common traps
- Subtracting instead of adding: (7 − 3) ÷ 2 = 2 is half the horizontal gap, not the midpoint's x-coordinate.
- Pairing an x-coordinate with a y-coordinate, such as averaging 3 with 4.
The midpoint is the working step at 18 Sep 2025, 09:00, Quant Q.18, where D, the midpoint of BC, is found from the centroid, and at 17 Sep 2025, 16:00, Quant Q.19, where the perpendicular bisector of (2, 8) and (6, 4) passes through their midpoint (4, 6).
Related PYQs
No directly related past PYQ was found.