Question

Co-ordinates of vertices of a triangle are $(0,0),(0,21)$ and $(21,0)$. How many points with integral coordinates lie inside this triangle?

• A. $190$
• B. $250$
• C. $200$
• D. $201$

Answer 1

Answer: (A)

$190$

The line joining the given points are $x=0,$ $y=0,$ $x+y=21$

Any point lying inside the triangle formed by these lines must follow the below conditions :

$x>0,$ $y>0,$ $x+y-21<0$

Let $x=1$ $\Rightarrow 0<y<20$ i.e, $19$ points

$x=2$ $\Rightarrow 0<y<19$ i.e, $18$ points

$x=19$ $\Rightarrow 0<y<2$ i.e, $1$ points

$\therefore$ Total $=19+18+........+1$

$=\frac{19\times 20}{2}$

$\Rightarrow$ Total $=190$

Hence, the answer is $190.$