Question

Suppose a triangle is formed by \(x+y=10\) and the coordinate axes. Then the number of points \((x,y)\), where \(x\) and \(y\) are natural numbers, lying inside the triangle is

• A. \(36\)
• B. \(55\)
• C. \(45\)
• D. \(30\)

Hint:

For lattice points inside a triangle bounded by coordinate axes and \(x+y=n\), use the condition \(x\gt 0,\ y\gt 0,\ x+y\lt n\).

Answer 1

The Correct Option is A

Step 1: Understand the triangle.
The triangle is formed by the line \[ x+y=10 \] and the coordinate axes.
So, the triangle lies in the first quadrant and is bounded by \[ x=0,\qquad y=0,\qquad x+y=10 \]

Step 2: Condition for points inside the triangle.
A point \((x,y)\) lies inside the triangle if \[ x\gt 0,\qquad y\gt 0,\qquad x+y\lt 10 \] Since \(x\) and \(y\) are natural numbers, we have \[ x=1,2,3,\ldots \] and \[ y=1,2,3,\ldots \]

Step 3: Count all possible points.
For each value of \(x\), count possible natural number values of \(y\).
If \(x=1\), then \[ y\lt 9 \] So, \[ y=1,2,\ldots,8 \] Number of values \(=8\).
If \(x=2\), then \[ y\lt 8 \] Number of values \(=7\).
Continuing in this way, we get \[ 8+7+6+5+4+3+2+1 \]

Step 4: Add the values.
\[ 8+7+6+5+4+3+2+1=36 \]

Step 5: Final conclusion.
Therefore, the number of points \((x,y)\), where \(x\) and \(y\) are natural numbers, lying inside the triangle is \[ \boxed{36} \]