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# Geometry

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How many lattice points (points with integer coordinates) are on the line segment whose endpoints are (3, 17)  and (81, 131)  (Include both endpoints in your count.).

I know that the slope is 19/13, but what do I do next?

Jul 23, 2024

#2
+1657
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Hi, AUnVerifiedTaxPayer! I will give this problem a shot!

So a lattice point is essentially any point on the graph. (1, 1), (2, 2) ,(3, 3) are ALL latice points!

However, we need to find ones on the line connecting  (3, 17)  and (81, 131)

To do this, let's first find the slope of the line. We have

$$\frac{131-17}{81-3}=\frac{114}{78} = \frac{19}{13}$$, so you are correct about the slope.

Now, one way you can do this is by just counting. Draw a graph, and use the slope to go up 19 and to the right 13 starting from (3, 17), eventually getting the next point. We count $$(3,17),(16,36),(29,55),(42,74),(55,93),(68,112),(81,131)$$

However, I will now do this another way.

Let's do this a simpler way rather than using the slope.

The slope​ indicates that for every increase of 13 units in the x-direction, the line moves 19 units in the y-direction.

The number of steps in the x-direction is given by the equation $$\text{Number of steps}= \frac{\text{total change in x}}{{\text{step size in x}}}$$

Now, from the slope, we know that the step size is 13.

Since we go from 3 to 81, the total change in x is 78.

Thus, we have 78/13 = 6.

Now, we are not done yet. We haven't included the start point, so adding 1, we have 6+1 = 7.

So 7 is our answer. I hope I did it correctly!

Thanks! :)

Jul 23, 2024
edited by NotThatSmart  Jul 23, 2024

#1
+1749
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We can solve this problem by finding the greatest common divisor (GCD) of the change in x and the change in y going from (3,17) to (81,131).

Steps to solve:

Find the change in coordinates:

Change in x (dx) = 81 - 3 = 78

Change in y (dy) = 131 - 17 = 114

Find the greatest common divisor (GCD) of dx and dy:

GCD(78, 114) = 6

Reduce the change in coordinates by the GCD:

dx' = dx / GCD = 78 / 6 = 13

dy' = dy / GCD = 114 / 6 = 19

The number of lattice points is the number of steps to move from (0, 0) to (dx', dy') along the lattice lines. This is essentially the sum of the absolute values of dx' and dy'.

Number of steps = |dx'| + |dy'| = 13 + 19 = 32

Add 1 to include both the starting and ending points:

Number of lattice points = 32 + 1 = 33

Therefore, there are 33 lattice points on the line segment connecting (3, 17) and (81, 131), including both endpoints.

Jul 23, 2024
#2
+1657
+4

Hi, AUnVerifiedTaxPayer! I will give this problem a shot!

So a lattice point is essentially any point on the graph. (1, 1), (2, 2) ,(3, 3) are ALL latice points!

However, we need to find ones on the line connecting  (3, 17)  and (81, 131)

To do this, let's first find the slope of the line. We have

$$\frac{131-17}{81-3}=\frac{114}{78} = \frac{19}{13}$$, so you are correct about the slope.

Now, one way you can do this is by just counting. Draw a graph, and use the slope to go up 19 and to the right 13 starting from (3, 17), eventually getting the next point. We count $$(3,17),(16,36),(29,55),(42,74),(55,93),(68,112),(81,131)$$

However, I will now do this another way.

Let's do this a simpler way rather than using the slope.

The slope​ indicates that for every increase of 13 units in the x-direction, the line moves 19 units in the y-direction.

The number of steps in the x-direction is given by the equation $$\text{Number of steps}= \frac{\text{total change in x}}{{\text{step size in x}}}$$

Now, from the slope, we know that the step size is 13.

Since we go from 3 to 81, the total change in x is 78.

Thus, we have 78/13 = 6.

Now, we are not done yet. We haven't included the start point, so adding 1, we have 6+1 = 7.

So 7 is our answer. I hope I did it correctly!

Thanks! :)

NotThatSmart Jul 23, 2024
edited by NotThatSmart  Jul 23, 2024