Question

ABCD is a quadrilateral such that AD = 9 cm, BC = 13 cm and $\angle DAB = \angle BCD = 90^\circ$. 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?

Accepted Answer

D. Let $DQ = x$, then $BP = 2x$. Area of $	riangle BPD = \frac{1}{2} \cdot AD \cdot BP = \frac{1}{2} \cdot 9 \cdot 2x = 9x$. Area of $	riangle BQD = \frac{1}{2} \cdot BC \cdot DQ = \frac{1}{2} \cdot 13 \cdot x = 6.5x$. Area of $PBQD = 9x + 6.5x = 15.5x \le 150$. $$15.5x \le 150 \implies x \le \frac{150}{15.5} \approx 9.67$$ Since $DQ = x$ must be a positive integer, $x$ can take values $1, 2, 3, \dots, 9$ (9 values).

Geometry: Maximum Area of Quadrilateral

Explanation

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