A quadratic equation

Question

Question: ABCD is a square of side 8cm. P and Q are points on AB and BC respectively such that AP=QC= χ cm. If the area of triangle ADP is two-thirds of the area of triangle BPQ, find the value of χ correct to 3 sig. fig.

I have found the area of triangle ADP =(BD•AP)÷2= 8χ÷2=4χ

How about triangle BPQ? One side is 8-χ But I can't find the height Or my calculation is wrong?

Answer 1

From the given informations we get the equation

$\dfrac{8\chi}{2}=\dfrac{2}{3}\cdot \dfrac{(8-\chi)^2}{2}$

which simplifies to

$\chi ^2-28 \chi +64=0$

$\chi=2 \left(7\pm\sqrt{33}\right)$

$\chi_1=2 \left(7-\sqrt{33}\right)\approx 2.51$

$\chi_2>8$ so we reject it

Answer 2

for the are of $$\Delta APD$$ we get $$A_{APD}=\frac{1}{2}x\cdot 8$$ and for area of triangle $$\Delta PBQ$$ we get $$A_{PBQ}=\frac{1}{2}(8-x)(8-x)$$ can you finish now?