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?

Well, to find the number of lattice points within the triangle, we can use a classic clown technique called counting on our fingers. Just kidding, I don't have fingers!

To tackle this problem, we can use a little bit of geometry combined with a sprinkle of humor. Let's start by drawing the triangle and connecting the points (1, 1), (9, 1), and (9, n).

Once we have the triangle on paper (or in our imagination), we can notice that the base of the triangle is made up of 8 lattice points (from x = 1 to x = 9). The vertical side of the triangle has a height of n - 1 because the point (9, 1) is included.

To determine the number of lattice points strictly inside the triangle, we can use a formula called Pick's Theorem. According to Pick's Theorem, the formula for the number of lattice points inside a polygon is:

A = i + (b/2) - 1

Where A represents the area of the polygon, i represents the number of interior lattice points, and b represents the number of lattice points on the boundary of the polygon.

In this case, the area of the triangle is (9-1)*(n-1)/2 = (8n-8)/2 = 4(n-1). So we can substitute that into the formula:

4(n - 1) = i + 8/2 - 1
4n - 4 = i + 4 - 1
4n - 4 = i + 3

To have exactly 560 lattice points strictly inside the triangle, we need to solve the equation 4n - 4 = 560 - 3, which simplifies to:

4n - 4 = 557
4n = 561
n = 561/4

So the integer value of n that satisfies the conditions is n = 140.25. But since we are dealing with lattice points (integer values), we round down and get n = 140.

Therefore, for n = 140, there are exactly 560 lattice points strictly in the interior of the triangle formed by the points (1, 1), (9, 1), and (9, n).

First, let's visualize the given points to better understand the problem:

Point A: (1, 1)
Point B: (9, 1)
Point C: (9, n)

Let's calculate the slope of line AB:

m_AB = (y2 - y1) / (x2 - x1)
= (1 - 1) / (9 - 1)
= 0 / 8
= 0

Since the slope is 0, line AB is a horizontal line.
Next, let's calculate the slope of line BC:

m_BC = (y2 - y1) / (x2 - x1)
= (n - 1) / (9 - 9)
= (n - 1) / 0 (undefined)

Since the slope is undefined, line BC is a vertical line.
Now, let's calculate the slope of line AC:

m_AC = (y2 - y1) / (x2 - x1)
= (n - 1) / (9 - 1)
= (n - 1) / 8

Since this is the slope for the diagonal line AC, let's analyze the conditions for a lattice point to be strictly in the interior of the triangle:

- The x-coordinate of the lattice point should be between 1 and 9.
- The y-coordinate of the lattice point should be between 1 and n.
- The slope of the line connecting the lattice point with point A should be greater than the slope of line AB.
- The slope of the line connecting the lattice point with point B should be between the slopes of lines AB and BC.
- The slope of the line connecting the lattice point with point C should be less than the slope of line AC.

Considering these conditions, let's solve the problem step by step.

1. The x-coordinate of the lattice point should be between 1 and 9.
Since we have only one possibility for the x-coordinate (i.e., 9), we can conclude that the x-coordinate of the lattice point is 9.

2. The y-coordinate of the lattice point should be between 1 and n.
Since we are given that there are exactly 560 lattice points strictly in the interior, the y-coordinate can't be smaller than 1. Thus, the y-coordinate of the lattice point should be between 1 and n, inclusive.

3. The slope of the line connecting the lattice point with point A should be greater than the slope of line AB (which is 0).
Let's calculate the slope between the lattice point and A:

m_lattice_A = (y_lattice - y_A) / (x_lattice - x_A)
= (y_lattice - 1) / (9 - 1)
= (y_lattice - 1) / 8

Since this slope should be greater than 0:

(y_lattice - 1) / 8 > 0

To maintain strict inequality, we can multiply both sides by 8:

y_lattice - 1 > 0

Add 1 to both sides:

y_lattice > 1

Therefore, the y-coordinate of the lattice point should be strictly greater than 1.

4. The slope of the line connecting the lattice point with point B should be between the slopes of lines AB (0) and BC (undefined).

Let's calculate the slope between the lattice point and B:

m_lattice_B = (y_lattice - y_B) / (x_lattice - x_B)
= (y_lattice - 1) / (9 - 9)
= (y_lattice - 1) / 0 (undefined)

Since this slope should be undefined and simultaneously greater than 0, there is no feasible value for the y-coordinate of the lattice point.

Therefore, we have determined that there are no lattice points strictly in the interior of the triangle for any integer value of n that satisfies the given conditions.

To find the integer value of n for which there are exactly 560 lattice points strictly in the interior of the triangle formed by the points (1, 1), (9, 1), and (9, n), we need to understand the geometric properties of the triangle and how to count the lattice points inside it.

First, let's consider the base of the triangle, which lies on the x-axis and runs from (1, 1) to (9, 1). There are 9 lattice points on this base, including both endpoints.

Next, we need to consider the line segment from (9, 1) to (9, n), which forms one of the sides of the triangle. The number of lattice points on this line segment can be found by counting the number of integers between 1 and n, inclusive. Since the y-coordinate varies from 1 to n, there are n-1 lattice points on this side.

Now, let's think about the height of the triangle, which is the perpendicular distance from the base to the top vertex (9, n). Since the base lies on the x-axis, the height is equal to the y-coordinate of the top vertex, which is n.

To count the lattice points strictly in the interior of the triangle, we can subtract the lattice points on the boundary (base and sides) from the total number of lattice points within the rectangle that encloses the triangle.

The rectangle can be formed by considering the coordinates of the two opposite corners: (1, 1) and (9, n). So the total number of lattice points within the rectangle is (9 - 1 + 1) * (n - 1 + 1) = 9n.

Now, we need to subtract the lattice points on the base, which is 9, and the lattice points on the side, which is n-1.

So, the number of lattice points strictly in the interior of the triangle is given by:
Interior points = Total points - Points on base - Points on side
= 9n - 9 - (n - 1)

According to the problem statement, this number is equal to 560:
9n - 9 - (n - 1) = 560

Simplifying the equation, we get:
8n - 8 = 560

Adding 8 to both sides:
8n = 568

Dividing both sides by 8:
n = 71

Therefore, the integer value of n that satisfies the condition is 71.