Suppose a triangle is formed by x + y = 10 and the coordinate axes. Then the number of points x , y where x…

Plotting the diagram of the given value we have,

Triangle formed will have vertices as

$O\left(0,0\right), A\left(10,0\right)$ and $B\left(0,10\right)$

If $x=1$, Point on the hypotenuse $\equiv \left(1,9\right)$

So, number of integral points lying inside triangle when $x=1$ is $8$ i.e $\left\{\left(1,1\right),\left(1,2\right)\cdots \left(1,8\right)\right\}$

Similarly,

when $x=2$, number of integral points lying inside triangle is $7$ i.e $\left\{\left(2,1\right),\left(2,1\right)\cdots \left(2,7\right)\right\}$

$\therefore$ Total number of integral points lying inside triangle are $=8+7+6+\dots +1$$=\frac{8\times 9}{2}=36$