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.

Explanation

In `DeltaABC`
`" "angleB= 90^(@)`
`therefore" "AB^(2)+BC^(2)=AC^(2)`
`rArr(p-4)^(2)+(3-7)^(2)+(p-7)^(2)+(3-3)^(2)=(7-4)^(2)+(3-7)^(2)`
`rArr" "p^(2)-8p+16+p^(2)-14p+49+0=9`
`rArr" "2p^(2)-22p+56=0`
`rArr" "p^(2)-11p+28=0`
`rArr" "p^(2)-7p-4p+28=0`
`rArr" "p(p-7)-4(p-7)=0`
`rArr" "(p-7)(p-4)=0`
`rArr" "p-7=0 or p-4=0`
`rArr" "p=7 or p=4`
when p=7, then the points B and C coincide and so no triangle is formed.
`therefore" "p ne7`
Hence, `p` =4