Question

If in a right angled triangle $ABC$, the hypotenuse $AB=p$, then $\overline{AB}.\overline{AC}+\overline{BC}.\overline{BA}+\overline{CA}.\overline{CB}=$ ?

• A. $p$
• B. $p^2$
• C. $p^3$
• D. $p^4$

Answer 1

The correct option is B $p^2$

We know that

$a^2+b^2=p^2$

$\overrightarrow{AB}.\overrightarrow{AC}+\overrightarrow{BC}.\overrightarrow{BA}+\overrightarrow{AC}.\overrightarrow{CB}=0$

$\overrightarrow{CA}.\overrightarrow{CB}=0$

$\overrightarrow{AB}\perp \overrightarrow{AC}$

$\left|\overrightarrow{AB}\right|\left|\overrightarrow{AC}\right|\cos \theta +\left|\overrightarrow{BC}\right|\left|\overrightarrow{BA}\right|\cos \left(90-\theta \right)+0$

$pb\cos \theta +ap\cos \left(90-\theta \right)$

$pb\cos \theta +ap\sin \theta$

$p\left(b\cos \theta +a\sin \theta \right)$

$p\left(p\cos \theta cos\theta +p\sin \theta \sin \theta \right)$

$=p^2$

Hence. the option $B$ is the correct answer.