3
The midpoint of a segment and finding a missing vertex
Mathematics Prep 1 Term 1, Unit 4, Lessons 5–6
GoalFind a midpoint from coordinates, and find a missing end point or vertex.
New words
midpoint · the point that divides a segment into two equal partsend point · one of the two points that bound a segmentaverage · the sum of two numbers divided by 2vertex · a corner point of a polygon
Explanation
The midpoint M of the segment from A(x₁, y₁) to B(x₂, y₂) is the average of the coordinates: M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2). It is an average, not a difference. If M and one end A are known, the other end is found by solving: (x₁ + x) ÷ 2 = xₘ, so x = 2xₘ − x₁, and the same for y. The diagonals of a parallelogram bisect each other, so both diagonals have the same midpoint. This is how to find a missing vertex: find the midpoint of one diagonal, then find the end point of the other diagonal.
ƒFormula
M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)
Worked examples
Midpoint. A(2, 6) and B(8, 2). M = ((2 + 8) ÷ 2, (6 + 2) ÷ 2) = (5, 4).
With negatives. A(−4, 3) and B(6, −5). x: (−4 + 6) ÷ 2 = 1. y: (3 + (−5)) ÷ 2 = −1. M = (1, −1).
Missing end point. M(3, 1) is the midpoint of AB and A is (1, 4). For x: (1 + x) ÷ 2 = 3, so x = 5. For y: (4 + y) ÷ 2 = 1, so y = −2. B is (5, −2).
Missing vertex. ABCD is a parallelogram with A(1, 1), B(5, 2) and C(7, 5). The midpoint of diagonal AC is (4, 3). It is also the midpoint of BD, so (5 + x) ÷ 2 = 4 gives x = 3 and (2 + y) ÷ 2 = 3 gives y = 4. D is (3, 4). Check: AB goes 4 right and 1 up, and DC goes 7 − 3 = 4 right and 5 − 4 = 1 up.
Class activity
Folding paper. With a grown-up, mark two points on squared paper and join them. Fold the paper so that the two points land on each other. The fold crosses the segment at its midpoint. Check it against the formula.
Practice
1
Find the midpoint of A(−2, 5) and B(4, 1).
((−2 + 4) ÷ 2, (5 + 1) ÷ 2) = (1, 3).
2
M(−1, 2) is the midpoint of AB and A is (3, −2). Find B.
x = 2 × (−1) − 3 = −5 and y = 2 × 2 − (−2) = 6. B is (−5, 6). Check: ((3 − 5) ÷ 2, (−2 + 6) ÷ 2) = (−1, 2).
3
ABCD is a parallelogram with A(0, 0), B(6, 1) and C(8, 4). Find D.
The midpoint of AC is (4, 2). Then D = (2 × 4 − 6, 2 × 2 − 1) = (2, 3).
4
M(5, 4) is the midpoint of A(k, 2) and B(8, 6). Find k.
(k + 8) ÷ 2 = 5, so k + 8 = 10 and k = 2. (Check y: (2 + 6) ÷ 2 = 4.)
5
Nour says the midpoint of (2, 6) and (8, 2) is (6, 4). What did she do wrong?
She subtracted (8 − 2 = 6, 6 − 2 = 4). The midpoint is the average: (5, 4).