The midpoints of the sides of triangle ABC are the points D, E, and F. Given that the coordinates of D, E, and F are (5, 1), (6, 5), and (0, 3), respectively, find the coordinates of A, B and C.
Step-by-step solution
To find: A, B and C
Idea: The midpoint rule turns into sums: B + C = 2D, C + A = 2E, A + B = 2F (coordinate by coordinate). Adding all three gives A + B + C, and subtracting each pair sum leaves one vertex.
- Let A (x1, y1), B (x2, y2), C (x3, y3). Midpoint rule (each coordinate of a midpoint is the average):
D of BC: x2 + x3 = 10, y2 + y3 = 2
E of CA: x3 + x1 = 12, y3 + y1 = 10
F of AB: x1 + x2 = 0, y1 + y2 = 61 mark - Add the three x-equations: 2(x1 + x2 + x3) = 22, so x1 + x2 + x3 = 11. Add the y-equations: 2(y1 + y2 + y3) = 18, so y1 + y2 + y3 = 9.1 mark
- A: x1 = 11 − 10 = 1, y1 = 9 − 2 = 7, so A (1, 7).½ mark
- B: x2 = 11 − 12 = −1, y2 = 9 − 10 = −1, so B (−1, −1).½ mark
- C: x3 = 11 − 0 = 11, y3 = 9 − 6 = 3, so C (11, 3).½ mark
- Check: midpoint of BC = ((−1 + 11) ÷ 2, (−1 + 3) ÷ 2) = (5, 1) = D ✓; of CA = ((11 + 1) ÷ 2, (3 + 7) ÷ 2) = (6, 5) = E ✓; of AB = ((1 − 1) ÷ 2, (7 − 1) ÷ 2) = (0, 3) = F ✓.½ mark
- (If the midpoints were matched to the sides differently, the same three points would come out, only with the letters A, B, C swapped.)
Check: Each vertex = sum of the two midpoints next to it − the opposite midpoint: A = E + F − D = (6 + 0 − 5, 5 + 3 − 1) = (1, 7) ✓.
Answer to write in the exam
Let A (x1, y1), B (x2, y2), C (x3, y3); D, E, F midpoints of BC, CA, AB
x2 + x3 = 10, x3 + x1 = 12, x1 + x2 = 0 ⇒ x1 + x2 + x3 = 11
y2 + y3 = 2, y3 + y1 = 10, y1 + y2 = 6 ⇒ y1 + y2 + y3 = 9
x1 = 11 − 10 = 1, y1 = 9 − 2 = 7
x2 = 11 − 12 = −1, y2 = 9 − 10 = −1
x3 = 11 − 0 = 11, y3 = 9 − 6 = 3
∴ A (1, 7), B (−1, −1), C (11, 3)
Common mistakes that cost marks
- Forgetting the factor 2: writing x2 + x3 = 5 instead of 10.
- Treating D, E, F as the vertices and finding the midpoints of DEF instead.
- Pairing the midpoints with the wrong sides in different equations. Fix once which side each midpoint is on and keep it.
How this can come in the exam
The midpoints of the sides of a triangle are (1, 2), (3, 4) and (5, 0). The sum of the x-coordinates of its vertices is
- 9
- 4.5
- 18
- 3
Show answer
(A) 9
Adding the three midpoint equations: 2 × (sum of vertex x-coordinates) = 2 × (1 + 3 + 5), so the sum is 9.
In triangle ABC, A = (2, 3). The midpoints of AB and AC are (5, 4) and (1, 6). Find B, C and the midpoint of BC.
Show answer
B: (2 + x) ÷ 2 = 5, (3 + y) ÷ 2 = 4 ⇒ B (8, 5) (1 mark). C: (2 + x) ÷ 2 = 1, (3 + y) ÷ 2 = 6 ⇒ C (0, 9) (1 mark). Midpoint of BC = (4, 7) (1 mark).Try one yourself
The midpoints of BC, CA and AB of triangle ABC are D (3, 2), E (−1, 4) and F (1, −2). Find A, B and C.
Show answer
D + E + F = (3, 4). A = (3, 4) − 2D = (−3, 0); B = (3, 4) − 2E = (5, −4); C = (3, 4) − 2F = (1, 8).
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