Animated Solution for Mathematics - Straight Lines: The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices $(0,0),(0,21)$ and $(21,0)$, is

Welcome, future engineer. Today, we are not just solving a coordinate geometry problem; we are embarking on a journey into the heart of discrete mathematics. When you look at a triangle with vertices at $(0,0)$, $(0,21)$, and $(21,0)$, you might see a simple shape.

But I want you to see something else. I want you to see a battlefield of points, a grid of possibilities waiting to be counted. The problem asks us to find the number of integral points—points where both $x$ and $y$ are integers—that lie strictly inside this triangle.

This is a classic JEE Advanced challenge, and it requires us to move beyond brute force and embrace the elegance of symmetry.

First, let us define our territory. We are in the first quadrant with vertices at $O(0,0)$, $A(0,21)$, and $B(21,0)$. The hypotenuse is the line segment connecting $(0,21)$ and $(21,0)$.

Using the intercept form of a line, $\frac{x}{a}+\frac{y}{b}=1$, we immediately see that our hypotenuse is defined by the equation:

For a point $(x,y)$ to be strictly inside the triangle, it must satisfy two conditions: it must be in the first quadrant ($x>0,y>0$) and it must lie below the hypotenuse ($x+y<21$). If we were to try and count these points one by one, we would be summing $20+19+18+\cdots +1$.

Imagine we complete this triangle into a square. We extend our boundaries to create a square with vertices at $(0,0)$, $(21,0)$, $(21,21)$, and $(0,21)$. This square is our playground, and it is perfectly symmetric about the diagonal $x+y=21$.

Let us calculate the total number of integral points strictly inside this square. The $x$-coordinates can range from $1$ to $20$, and the $y$-coordinates can also range from $1$ to $20$.

Since we are looking for points strictly inside, we exclude the boundaries where $x=0,x=21,y=0,$ or $y=21$. By the fundamental principle of counting, the total number of integral points inside this square is:

Now, we must address the diagonal. The line $x+y=21$ cuts through our square, and any point on this line is not strictly inside our triangle.

If $x=1$, then $y=20$. If $x=2$, then $y=19$. This continues until $x=20$, where $y=1$. These are the points $(1,20),(2,19),\dots ,(20,1)$.

Counting them, we find exactly $20$ points lying on the diagonal. We must remove these from our total count of $400$:

These $380$ points are the ones that are not on the diagonal. Because of the perfect symmetry of the square across the line $x+y=21$, exactly half of these points must lie below the diagonal (inside our triangle).

This is the moment of truth. We take our $380$ points and divide them by $2$:

There it is. $190$ integral points. We used the geometry of the square to simplify the problem into a subtraction and a division.

A General Formula

Before you go, I want to leave you with a powerful tool for your JEE arsenal. For any right-angled triangle with vertices at $(0,0)$, $(n,0)$, and $(0,n)$, the number of strictly interior integral points is given by the formula:

Let us verify this with our problem where $n=21$:

The formula holds perfectly. This is the beauty of mathematics—when you understand the underlying structure, you don't just solve problems; you master them.