question_answer

ABC is a right angled triangle, right angled at C and p is the length of the perpendicular from C on AB. If a, b and c are the sides of the triangle, then which one of the following is correct?

A)  \[({{a}^{2}}+{{b}^{2}}){{p}^{2}}={{a}^{2}}{{b}^{2}}\]

B)  \[{{a}^{2}}+{{b}^{2}}={{a}^{2}}{{b}^{2}}{{p}^{2}}\]

C) \[{{p}^{2}}={{a}^{2}}+{{b}^{2}}\]          

D)  \[{{p}^{2}}={{a}^{2}}-{{b}^{2}}\]

Correct Answer: A

Solution :

In right \[\Delta ABC,\]       Area \[=\frac{1}{2}\times a\times b\] Again, in right \[\Delta ABC,\] Area \[=\frac{1}{2}\times AB\times DC\] \[\Rightarrow \]   \[\frac{1}{2}ab=\frac{1}{2}\times c\times p\] \[\Rightarrow \]   \[ab=p(\sqrt{{{a}^{2}}+{{b}^{2}})}\]\[(\because {{c}^{2}}={{a}^{2}}+{{b}^{2}})\] \[\Rightarrow \]   \[{{a}^{2}}{{b}^{2}}={{p}^{2}}({{a}^{2}}+{{b}^{2}})\]