Question

Number of points with integral co-ordinates that lie inside a triangle whose co-ordinates are $(0,0)$ , $(0,21) and (21,0)

• A. $210$
• B. $190$
• C. $220$
• D. None

Answer 1

Answer: (C)

$220$

Let the vertices of triangle be $A(21,0):B(0,21)$ and $D(0,0)$

Thus any point in the interior of the triangle begin first quadrant

Therefore $a>0$ $b>0$

point (a, b) lies on the same side of AB where for $(0,0)$

$x+y-21=0$

$-21<0$

$\therefore a+b-21<0$

$a+b<21$

for $a=1$ $b<21-1$ i.e., $b<20$; $b\in (1,19)$

Total $19$ values of b

for $a=2$ $b<21-2$ i.e., $b\in (1,18)$

$\therefore$ no. of integral points $19+18+.....+1$

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

$=190$ integral points.