# Different Value of Side $BC$ by Similarity and Pythagorean Theorem

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?

• There is no correct answer. The given information is inconsistent.
– Blue
Sep 3, 2020 at 6:55
• You are getting two different answers as the dimensions given are incorrect for a right angled triangle. Sep 3, 2020 at 7:01
• Thanks. When I constructed triangel ABC with BC=8.1,I found CD 6.5cm and when I constructed ABC with BC=6.6cm, I found CD=4.9cm. Sep 3, 2020 at 7:16

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 $$\sqrt{5.7^2-3.8^2}\neq\frac{361}{135}$$
• But in the $\triangle$ABD and $\triangle$BCD , only one angle is equal to other triangle i.e. 90°. Sep 3, 2020 at 8:12
• @Mayank Also, $\measuredangle BAD=\measuredangle CBD.$ Sep 3, 2020 at 8:15