Question

a triangle is formed drawn on the xy plane by three vertices (0 20) (0 0) and (20 0). the number of points with integer coordinates inside the triangle (excluding all the points on the boundary) are?

Answer 1

The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) are 171

Explanation

The vertices of the triangle are: A (0, 20) , O(0, 0) and B(20,0)

The formula of finding the number of points with integer co ordinates inside a triangle is :

$I= A -\frac{B}{2} +1$ , where A= Area of the triangle and B = total number of border points

First, we will find the Area of the triangle using the formula:

$Area = \frac{1}{2} * base * height\\ \\ A= \frac{1}{2} * 20 *20 \\ \\ A = 200$

Now except the 3 vertices,

Number of border points with integer co ordinate on line OA = 19,

Number of border points with integer co ordinate on line OB = 19 and

Number of border points with integer co ordinate on line AB = 19.

So, the total border points, $B =3+19+19+19= 60$

According to the above formula,

$I= A - \frac{B}{2}+1\\ \\ I= 200 - \frac{60}{2} +1\\ \\ I= 200-30+1 = 171$

So, the number of points with integer coordinates inside the triangle (excluding all the points on the boundary) are 171.