Triangle is formed by the co-ordinates (0,0),(0,21) and (21,0). Find the number of integral co ordinate strictly inside the triangle (integral co ordinates of both x and y)

Options

A. 190
B. 105
C. 231
D. 205

Answer

To find the number of integral coordinates strictly inside the triangle formed by the points (0,0), (0,21), and (21,0), we can follow these steps:

### Step 1: Understand the Triangle
The vertices of the triangle are:
- A(0, 0)
- B(0, 21)
- C(21, 0)

The triangle lies in the first quadrant of the Cartesian plane.

### Step 2: Determine the Equation of the Line
The line connecting points B and C can be expressed with the equation:
\[ x + y = 21 \]
This line forms the hypotenuse of the triangle.

### Step 3: Count Integral Points along the Line
To find the number of integral points strictly inside the triangle, we need to consider the integral points along the line segment BC for various values of x.

1. For \( x = 1 \):
- The equation becomes \( 1 + y = 21 \) → \( y = 20 \)
- Integral points on the line from (1, 20) to (0, 21) = 19 points (from y=1 to y=20).

2. For \( x = 2 \):
- The equation becomes \( 2 + y = 21 \) → \( y = 19 \)
- Integral points on the line from (2, 19) to (0, 21) = 18 points.

3. For \( x = 3 \):
- The equation becomes \( 3 + y = 21 \) → \( y = 18 \)
- Integral points on the line from (3, 18) to (0, 21) = 17 points.

### Step 4: Continue Counting Points
Continuing this process, we can see that for each increase in x, the number of integral points decreases by 1.

- For \( x = 4 \): 16 points
- For \( x = 5 \): 15 points
- ...
- For \( x = 19 \): 1 point

### Step 5: Sum the Integral Points
The total number of integral points can be calculated by summing the series:
\[ 19 + 18 + 17 + ... + 1 \]

This is an arithmetic series where:
- First term \( a = 1 \)
- Last term \( l = 19 \)
- Number of terms \( n = 19 \)

The sum \( S \) of the first n natural numbers is given by:
\[ S = \frac{n}{2} \times (a + l) \]
\[ S = \frac{19}{2} \times (1 + 19) = \frac{19}{2} \times 20 = 19 \times 10 = 190 \]

### Final Answer
The number of integral coordinates strictly inside the triangle is **190**.

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