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?
\(24\ \text{units}\)
Let the side of square PQRS be \(a\), with \(P(0,0), Q(a,0), R(a,a), S(0,a)\). Since \(PM:MS=2:1\) on \(PS\), \(M=\left(0,\tfrac{2a}{3}\right)\). Since \(SN:NR=1:2\) on \(SR\), \(N=\left(\tfrac{a}{3},a\right)\). Computing the area of triangle \(MQN\) gives \(\text{Area}=\dfrac{5a^2}{18}\). Setting this equal to 10 gives \(a^2=36\Rightarrow a=6\). Perimeter \(=4a=24\) units.
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