The number of points, having both co-ordinates as integers, that lie in the interior of the triangle

Q: Why do we subtract the points on the diagonal? Shouldn't they be included?

Great question! The problem specifically asks for points in the *interior* of the triangle. The line $x + y = 41$ forms the hypotenuse, which is the boundary of the triangle. Points on the boundary are not in the interior, so we must exclude them.

Q: How do we know there are exactly 40 points on the diagonal $x + y = 41$?

Think about the constraints. For $x + y = 41$ with $x, y > 0$, $x$ can take any integer value from $1$ to $40$. For each $x$, $y$ is uniquely determined as $41 - x$. Since $x$ ranges from $1$ to $40$, there are exactly $40$ such pairs $(x, y)$.

Q: Is there a faster way to do this without the square method?

Yes, you can use the general formula for the number of interior lattice points in a right triangle with vertices $(0,0), (n,0), (0,n)$, which is $\frac{(n-1)(n-2)}{2}$. For $n=41$, this gives $\frac{40 \times 39}{2} = 780$. Both methods are powerful, but the square method helps build geometric intuition!

Question

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0, 0), (0, 41)$ and $(41, 0)$ is

Options:

(A) 820

(B) 780

(C) 901

(D) 861

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a coordinate plane, looking at a right-angled triangle. Its vertices are anchored at the origin $(0,0)$, the point $(41,0)$ on the x-axis, and $(0,41)$ on the y-axis.

Our mission is to count every single point inside this triangle where both coordinates are integers. This is a classic problem involving the counting of lattice points.

### Defining the Boundaries

A point $(x,y)$ is strictly inside our triangle if it satisfies three conditions:
1. $x > 0$
2. $y > 0$
3. The point must lie below the hypotenuse.

The equation of the line connecting $(41,0)$ and $(0,41)$ is $x + y = 41$. Therefore, for any point to be in the interior, it must satisfy the inequality:
$$x + y < 41$$

### The Bounding Square Trick

If we try to count these points row by row, we might get lost in the arithmetic. Instead, let us use the technique of the Bounding Square.

Imagine a square with vertices $(0,0), (41,0), (41,41),$ and $(0,41)$. This square encapsulates our triangle. The interior of this square consists of all integer points where $1 \le x \le 40$ and $1 \le y \le 40$.

Calculating the total number of interior points in this square is straightforward: we have $40$ choices for $x$ and $40$ choices for $y$. This gives us:
$$40 	imes 40 = 1600 	ext{ points}$$

### The Diagonal and Symmetry

Now, consider the line $x + y = 41$ that cuts through our square. This line acts as a diagonal. Any integer point in the square either lies on this line, below it, or above it.

The points on the line are $(1, 40), (2, 39), \dots, (40, 1)$. Counting these, we find exactly $40$ points sitting on the boundary.

Since we only want the interior points of the triangle, we must exclude these $40$ points from our total of $1600$. This leaves us with:
$$1600 - 40 = 1560 	ext{ points}$$

### The Final Symmetry

The $1560$ remaining points are distributed symmetrically. Half of them lie below the line $x + y = 41$ (inside our triangle), and the other half lie above it.

By the principle of symmetry, we simply divide by two:
$$\frac{1560}{2} = 780$$

### The General Formula

For the curious mind, there is a general rule for any such triangle with vertices $(0,0), (n,0),$ and $(0,n)$. The number of interior lattice points is given by the formula:
$$\frac{(n-1)(n-2)}{2}$$

Plugging in $n=41$, we get:
$$\frac{40 	imes 39}{2} = 780$$

Whether you use the square method or the formula, the result is the same. The total number of interior lattice points is **780**.

Animated Solution for Mathematics - Straight Lines: The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0, 0), (0, 41)$ and $(41, 0)$ is

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Visualized Solution

$V i s u a l i z in g t h e T r ian g l e$

$D e f inin g I n t er i or C o n d i t i o n s$

$T h e B o u n d in g S q u a r e T r i c k$

$C o u n t in g S q u a r e I n t er i or P o in t s$

$P o in t s o n t h e D ia g o na l$

$S u b t r a c t in g D ia g o na l P o in t s$

$U s in g S y mm e t r y$

$G e n er a l F or m u l a$

The Sigma Insight: Various Forms of Equations of a Line

Analyzing the Setup

Defining the Boundaries

The Bounding Square Trick

The Diagonal and Symmetry

The Final Symmetry

The General Formula

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Visualizing the Triangle

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Defining Interior Conditions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Bounding Square Trick

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Counting Square Interior Points

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Points on the Diagonal

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Subtracting Diagonal Points

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Using Symmetry

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } General Formula