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A triangle in the coordinate plane is formed by the axes and the line 3x + 5y = 30.  How many points with integer coordinates lie inside the triangle?

 Jun 17, 2020
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First plot it out on the coordinate plane. The line 3x + 5y = 30 has the points (0, 6) and (10, 0). Because it's formed by the axes, it's a right triangle with height 6 and base 10. 

 

Now we can use Pick's Theorem

\(A=I+\frac{1}{2}B-1\)

where A=area, I= Number of lattice points in the interior, and B= Number of lattice points on boundary

 

We already know \(A=\frac{6*10}{2}=30\)

 

Now for finding B, the number of lattice points on the boundary.

 

We know the number of lattice points on the axis: 10+6+1 (for the origin) = 17. 

Now we have to find the lattice points on the hypotenuse of the triangle.

You can use this formula: For the line from (a,b) to (c,d), the number of lattice points is \(gcd(𝑐−𝑎,𝑑−𝑏)+1.\)

We plug our values in to find \(gcd(10,6)+1 = 2+1 = 3\). We have to subtract 2 because we already counted the x and y intercepts on the axis.

OR

use trial and error by putting in values for 0 < y < 6 and 0 < x < 10 into our equation 3x+5y=30. This only comes up with one value (5, 3).

So B=17+1=18

 

Plug our values in, solve for \(I\), and you have your answer.

\(30=I+\frac{1}{2}*18-1\)

 Jun 17, 2020

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