How many points with integer coordinates are in the triangle with vertices $(0, 0)$ , $(0, 21)$ and $(21, 0)$ , including the points on its boundary?

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How many points with integer coordinates are in the triangle with vertices $(0, 0)$ , $(0, 21)$ and $(21, 0)$ , including the points on its boundary?

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Answer 1 · 2018-01-06T22:54:41

$253$

Explanation:

The number of integral points on the diagonal line segment joining $(0, n)$ and $(n, 0)$ is $n+1$ , being $(0, n)$ , $(1, n-1)$ , $(2, n-2)$ ,..., $(n, 0)$ .

So there are $22$ integral points on the line segment $x+y = 21$ between $(0, 21)$ and $(21, 0)$ , then $21$ points on the next diagonal joining $(0, 20)$ and $(20, 0)$ , and so on down to one point at $(0, 0)$ .

So the total number of integral points in the triangle is:

$sum_(n=1)^22 n = 1/2 (color(blue)(22))(color(blue)(22)+1) = 253$

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How many points with integer coordinates are in the triangle with vertices $(0, 0)$ , $(0, 21)$ and $(21, 0)$ , including the points on its boundary?

1 Answer

Explanation:

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Science

Math

Humanities

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How many points with integer coordinates are in the triangle with vertices (0, 0)(0, 0), (0, 21)(0, 21) and (21, 0)(21, 0), including the points on its boundary?

1 Answer

Explanation:

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How many points with integer coordinates are in the triangle with vertices $(0, 0)$ , $(0, 21)$ and $(21, 0)$ , including the points on its boundary?