2026 · Q72
PQRS is a square. M is a point on PS such that PM : MS = 2 : 1 and N is a point on SR such that SN : NR = 1 : 2. If the area of triangle MQN is 10 square units, then what is the perimeter of the square?
Elementary MathematicsGeometry: Lines, Angles and Triangles
- A24 units
- B28 units
- C32 units
- D36 units
Explanation
Put P = (0, a), Q = (a, a), R = (a, 0), S = (0, 0) for a square of side a. Then M = (0, a/3) (PM : MS = 2 : 1) and N = (a/3, 0) (SN : NR = 1 : 2). By the shoelace formula, area(MQN) = ½|0·(a − 0) + a(0 − a/3) + (a/3)(a/3 − a)| = ½·5a²/9 = 5a²/18. Setting this to 10 gives a² = 36, so a = 6 and the perimeter is 24 units, option A.