Question

The co-ordinates of midpoints of the sides AB, BC, and CA of a triangle ABC are (3, 6), (5, 3), and (8, 5) respectively. The co-ordinates of vertex A are:

• A. (10, 2)
• B. (0, 4)
• C. (6, 8)
• D. (7, 9)

Answer 1

The Correct Option is C

Concept:

• The midpoint of a side of a triangle links the two vertices it joins through the midpoint formula.
• If D, E, F are midpoints of AB, BC, CA of triangle ABC, then vertex A = D + F - E (this comes directly from adding and subtracting the three midpoint equations).
• The same shortcut gives B and C from the other combinations.

Step 1: Assign the given midpoints
D = midpoint of AB = (3, 6)
E = midpoint of BC = (5, 3)
F = midpoint of CA = (8, 5)

Step 2: Apply the vertex formula A = D + F - E
x-coordinate of A = 3 + 8 - 5 = 6
y-coordinate of A = 6 + 5 - 3 = 8

Step 3: Verify the result using the midpoint definition
With A = (6, 8) and D = (3, 6) as midpoint of AB, B = (2 x 3 - 6, 2 x 6 - 8) = (0, 4)
With F = (8, 5) as midpoint of CA, C = (2 x 8 - 6, 2 x 5 - 8) = (10, 2)
Midpoint of B(0, 4) and C(10, 2) = (5, 3), which matches E, confirming A = (6, 8) is correct

Final Answer: (6, 8)