Question

In the figure, seg AB $\parallel$ seg DC. Using the information given in the figure, find the value of x.

Answer 1

$$\begin{gathered} \mathrm{Given}:\text{S}\mathrm{eg}\mathrm{AB}\parallel \text{S}\mathrm{eg}\mathrm{DC} \\ \text{In}△\mathrm{DOC}\text{and}△\mathrm{BOA}, \\ \angle \mathrm{DCO}=\angle \mathrm{BAO}[\mathrm{Alternate}\mathrm{angles}] \\ \angle \mathrm{CDO}=\angle \mathrm{ABO}[\mathrm{Alternate}\mathrm{angles}] \\ \mathrm{Therefore},\mathrm{by}\mathrm{AA}\mathrm{similarity}\mathrm{tes}\text{t,} \\ △\mathrm{DOC}~△\mathrm{BOA} \\ \mathrm{Hence},\frac{\mathrm{DO}}{\mathrm{BO}}=\frac{\mathrm{OC}}{\mathrm{OA}}=\frac{\mathrm{CD}}{\mathrm{AB}} \\ \frac{\mathrm{DO}}{\mathrm{BO}}=\frac{\mathrm{OC}}{\mathrm{OA}} \\ \Rightarrow \frac{3}{x-3}=\frac{x-5}{3x-19} \\ \mathrm{Cross}-\mathrm{multiplying},\text{we get:} \\ 3(3x-19)=(x-5)(x-3) \\ \Rightarrow 9x-57=x^2-3x-5x+15 \\ \Rightarrow 9x-57=x^2-8x+15 \\ \Rightarrow x^2-17x+72=0 \\ \Rightarrow x^2-9x-8x+72=0 \\ \Rightarrow x(x-9)-8(x-9)=0 \\ \Rightarrow (x-8)(x-9)=0 \\ \Rightarrow x=8\mathrm{or}x=9 \end{gathered}$$