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# Triangles

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A lattice point is an ordered pair (x, y) where both x and y are integers. A triangle is formed
by the three points (1, 1), (9, 1), and (9, n). For what integer value of n > 0 are there exactly 560 lattice
points strictly in the interior of the triangle?

Oct 26, 2022

#2
+118572
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The first think I would do is start drawing triangles on some grid paper.

They way maybe I can come up with some formula for this.

Have you tried that?   Or have you thought of anything else?

Nov 4, 2022
#3
+118572
+1

A lattice point is an ordered pair (x, y) where both x and y are integers. A triangle is formed
by the three points (1, 1), (9, 1), and (9, n). For what integer value of n > 0 are there exactly 560 lattice
points strictly in the interior of the triangle?

I'm quite sure there is a better way to do this

but

First I translated the points to (0,0), (8,0) and (8,n)  n>=0   because i thought it sould be easier

then  drew up a dot grid and counted the number of extra dots each time n increased by 1

this looks like a pattern, assuming that it continues this way ...

560/28 = 20

20+1=21

I think the answer is 21 ....

Nov 5, 2022
edited by Melody  Nov 5, 2022