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ABCD is a quadrilateral such that AD = 9 cm, BC = 13 cm and $$\angle$$DAB = $$\angle$$BCD = 90°. P and Q are two points on AB and CD respectively, such that DQ : BP = 1 : 2 and DQ is an integer. How many values can DQ take, for which the maximum possible area of the quadrilateral PBQD is 150 sq.cm?
Let $$DQ = x$$, => $$BP = 2x$$
Acc. to ques,
=> $$ar (\triangle BPD) + ar (\triangle BQD) \leq ar (PBQD)$$
=> $$(\frac{1}{2} \times AD \times BP) + (\frac{1}{2} \times BC \times QD) \leq 150$$
=> $$(\frac{1}{2} \times 9 \times 2x) + (\frac{1}{2} \times 13 \times x) \leq 150$$
=> $$31x \leq 300$$ => $$x \leq \frac{300}{31}$$
=> $$x \leq 9.68$$
Thus, for $$x$$ to be an integer and positive, 9 different values (1 to 9) are possible.
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