question_answer

The points \[A(4,7),\,\,B(p,3)\] and \[C(7,3)\] are the vertices of a right triangle, right-angled at B. Find the value of p.

Answer:

The given points are \[A(4,7),\,\,B(p,3)\]and \[C(7,3)\].

Since A, B and C are the vertices of a right angled triangle

Then,                \[{{(AB)}^{2}}+{{(BC)}^{2}}={{(AC)}^{2}}\]         [By Pythagoras theorem]

\[[{{(p-4)}^{2}}+{{(3-7)}^{2}}]+[{{(7-p)}^{2}}+{{(3-3)}^{2}}]=[{{(7-4)}^{2}}+{{(3-7)}^{2}}]\]

\[{{(p-4)}^{2}}+{{(-4)}^{2}}+{{(7-p)}^{2}}={{(3)}^{2}}+{{(-4)}^{2}}\]

\[{{p}^{2}}+16-8p+16+49+{{p}^{2}}-14p=9+16\]

\[2{{p}^{2}}-22p+56=0\]

\[{{p}^{2}}-11p+28=0\]

\[{{p}^{2}}-7p-4p+28=0\]

\[p(p-7)-4(p-7)=0\]

\[p=4\] or \[7\]

\[p\ne 7\] (As B and C will coincide)

So, \[p=4\].