Question :
In the given figure, $\angle ABC=90°$ and BD$\perp$AC. If $AB=5.7$cm, $BD=3.8$cm and $CD=5.4$cm, find $BC$.
By similarity in $\triangle ABC \sim \triangle BDC$, (By $\frac{AB}{BD}=\frac{BC}{DC}$ ) I got $BC=8.1$cm but by Pythagorean theorem I got $BC=6.6$.
Which one is right?
Because this triangle does not exist!
Indeed, by the Pythagorean theorem we obtain: $$AD=\sqrt{5.7^2-3.8^2}.$$
Since $$\measuredangle BAD=90^{\circ}-\measuredangle ABD=\measuredangle CBD,$$ we see that: $\Delta ABD\sim\Delta BCD$ and we obtain: $$BD^2=AD\cdot DC,$$ which gives $$AD=\frac{3.8^2}{5.4}=\frac{361}{135}$$ and easy to see that $$\sqrt{5.7^2-3.8^2}\neq\frac{361}{135}$$