Okay, let's tackle this problem step by step. The question is asking for the number of integer coordinate points strictly inside a given rectangular region. The vertices of the rectangle are (5,4), (-5,4), (-5,-4), and (5,-4). So first, I need to visualize this rectangle.

Hmm, the vertices are at (5,4), (-5,4), (-5,-4), and (5,-4). If I plot these points, the rectangle is centered at the origin (0,0). The length along the x-axis from -5 to 5 gives a width of 10 units, and the length along the y-axis from -4 to 4 gives a height of 8 units. So, the rectangle is 10 units wide and 8 units tall.

Now, the problem is to find all the integer coordinate points that are strictly inside this rectangle. That means the points should not lie on the boundary of the rectangle. So, I need to figure out how many such points there are.

First, let's recall that for a rectangle with integer coordinates, the number of interior points can be found using the formula: (width - 1) * (height - 1). Wait, but let me verify that. If the rectangle is from x = a to x = b and y = c to y = d, then the number of integer points strictly inside would be (b - a - 1) * (d - c - 1). Let me check with an example.

Suppose we have a rectangle from x = 1 to x = 3 and y = 2 to y = 4. The width is 3 - 1 = 2 units, but the number of interior points along the x-axis would be 3 - 1 - 1 = 1 (points at x=2). Similarly, the height is 4 - 2 = 2 units, so interior points along y-axis would be 4 - 2 -1 = 1 (y=3). Therefore, total interior points would be 1*1 =1, which is correct (the point (2,3)). So the formula seems to hold here.

Applying that to our problem: the rectangle goes from x = -5 to x =5, so the width is 5 - (-5) =10 units. The interior points along x would be 10 - 1 =9 points. Similarly, along y, from -4 to4, the height is 8 units, so interior points along y would be 8 -1 =7 points. Therefore, total interior points would be 9*7=63. Wait, but let me think again.

Wait, actually, the formula is (right - left -1) * (top - bottom -1). In our case, right is 5, left is -5, top is4, bottom is -4. So, right - left =5 - (-5)=10, so interior x points are 10 -1=9. Similarly, top - bottom=4 - (-4)=8, so interior y points=8 -1=7. Therefore, 9*7=63. So, 63 points. But let me verify this with another approach to be sure.

Alternatively, maybe I can count the number of integer points strictly inside the rectangle by considering the ranges of x and y coordinates. For the x-coordinate, strictly inside the rectangle means x must be greater than -5 and less than5. Since x must be an integer, the possible x values are -4, -3, ..., 4. That's from -4 to4 inclusive. How many integers is that? Let's calculate: from -4 to4, the number of integers is 4 - (-4) +1 =9. Wait, 4 - (-4) is 8, plus1 is9. So 9 x-coordinates. Similarly, for y-coordinate, strictly inside means y must be greater than -4 and less than4. So y can be -3, -2, ...,3. How many integers is that? From -3 to3 inclusive. 3 - (-3) +1=7. So 7 y-coordinates. Therefore, total points=9*7=63. That matches the previous result.

But wait, let me check with another method to make sure. Let's imagine the rectangle as a grid. The rectangle spans from x=-5 to5, so in terms of integer x-values, from -5 to5 inclusive. The interior points are those not on x=-5 or x=5. Similarly, for y, interior points are not on y=-4 or y=4.

So, for x, the integers from -5 to5 inclusive are 11 points (including both ends), but interior points would exclude x=-5 and x=5, so 11-2=9. Similarly, for y, integers from -4 to4 inclusive are9 points, but interior points exclude y=-4 and y=4, so9-2=7. Therefore, the total interior points are9*7=63. So, yes, that seems correct.

But let me make sure there isn't a mistake in the coordinates. The rectangle's vertices are (5,4), (-5,4), (-5,-4), (5,-4). So, the rectangle is indeed 10 units wide (from x=-5 to5) and 8 units tall (from y=-4 to4). So, if we consider all integer points where -5 <x <5 and -4 <y <4, then x can be from -4 to4 inclusive, which is9 values, and y can be from -3 to3 inclusive, which is7 values. Multiplying gives63.

Alternatively, if I consider the area of the rectangle, which is width*height=10*8=80. Then, the number of interior points should be area minus boundary points. But wait, that might not be straightforward because the area is in continuous terms, while the number of integer points is discrete. However, maybe we can use Pick's theorem here. Pick's theorem states that for a simple polygon with integer coordinates, the area A is equal to I + B/2 -1, where I is the number of interior integer points and B is the number of boundary integer points.

But in this case, the rectangle is a lattice polygon, so Pick's theorem applies. Let's try it.

First, compute the area A. As above, the area is10*8=80.

Next, compute B, the number of boundary points. For a rectangle, the boundary points are the points on the four edges. Each edge is a line segment between two vertices. For each edge, the number of integer points can be calculated by the gcd of the differences in coordinates plus1, but since the edges are axis-aligned, it's simpler.

For the top edge from (-5,4) to (5,4): it's a horizontal line at y=4 from x=-5 to5. The number of integer points here is5 - (-5) +1=11. Similarly, the bottom edge from (-5,-4) to (5,-4): also11 points.

The left edge from (-5,4) to (-5,-4): vertical line at x=-5 from y=4 to-4. The number of integer points here is4 - (-4) +1=9.

The right edge from (5,4) to (5,-4): vertical line at x=5, same as left edge,9 points.

So total boundary points B=11+11+9+9=40.

But wait, in Pick's theorem, B counts each vertex only once. However, in our case, the four vertices are each counted twice (once for each adjacent edge). So, the total boundary points according to the edges would be 11+11+9+9=40, but since each vertex is shared by two edges, the actual number of distinct boundary points is (11-2) + (9-2)*2 +4= Hmm, maybe this approach is complicating.

Alternatively, the number of boundary points for a rectangle with length L and width W (in terms of number of unit edges) is 2(L + W). But in our case, the rectangle is from x=-5 to5, which is10 units wide, and y=-4 to4, which is8 units tall. So, the number of boundary points would be2*(10 +8)=36. But wait, that's not matching the previous calculation.

Wait, maybe I need to clarify. The number of boundary points for a rectangle with integer vertices is calculated by considering each side. For the horizontal sides (top and bottom), the number of points is (5 - (-5) +1)=11 each, as we calculated before. For the vertical sides (left and right), the number is (4 - (-4) +1)=9 each. So total boundary points:11+11+9+9=40. However, each corner is counted twice in this total. There are four corners, so we need to subtract 4 to avoid double-counting. Therefore, B=40 -4=36. So, according to Pick's theorem, A = I + B/2 -1. Therefore, 80 = I + 36/2 -1 =>80 = I +18 -1 =>80=I +17 =>I=63. Which matches the previous result. Therefore, the number of interior points is63. So, this confirms the answer.

Therefore, the answer is63. But let me just double-check once more to be absolutely sure.

Another way: think of the rectangle as a grid from x=-5 to5 and y=-4 to4. The interior points must satisfy -5 <x <5 and -4 <y <4. So x can be -4,-3,-2,-1,0,1,2,3,4 (9 values) and y can be -3,-2,-1,0,1,2,3 (7 values). Therefore, 9*7=63. Yep, that's consistent.

Alternatively, if I list some points: for x=-4, y can be -3,-2,-1,0,1,2,3 (7 points). Similarly, x=-3: same y-values, etc., up to x=4. So 9 x-values, each with7 y-values. 9*7=63.

Yes, that all adds up. Therefore, I can be confident that the answer is63.

**Final Answer**
The number of integer coordinate points strictly inside the rectangular region is \boxed{63}.
Therefore, the number of points with integer coordinates that will be strictly in the interior of this rectangular region is $\boxed{63}$.
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