The problem involves a rectangle with sides parallel to the coordinate axes, positioned below the x-axis and to the right of the y-axis. The coordinates of points A, B, C, D are whole numbers. I need to find which point gives the smallest value when I divide the y-coordinate by the x-coordinate.

First, since the rectangle is below the x-axis and to the right of the y-axis, all points will have positive x-coordinates and negative y-coordinates because "below the x-axis" means negative y-values, and "to the right of the y-axis" means positive x-values.

Let me denote the coordinates:

• Let the x-coordinate of the left side be $a$, so points A and D have x-coordinate $a$.

• Let the x-coordinate of the right side be $b$, so points B and C have x-coordinate $b$.

• Since it's below the x-axis, y-coordinates are negative. Let the y-coordinate of the bottom side be $-c$, so points A and B have y-coordinate $-c$.

• Let the y-coordinate of the top side be $-d$, so points D and C have y-coordinate $-d$.

In a rectangle, opposite sides are equal, and since sides are parallel to axes, the differences in x and y should be consistent.

Assume A is at the bottom-left, B at bottom-right, C at top-right, D at top-left.

So:

• A: (x_A, y_A) = (a, -c)

• B: (x_B, y_B) = (b, -c)

• C: (x_C, y_C) = (b, -d)

• D: (x_D, y_D) = (a, -d)

Since it's a rectangle, the length of AB should equal CD, and AD should equal BC.

AB is horizontal, so length is |b - a|.

AD is vertical, so length is | -d - (-c) | = |c - d|.

Similarly for others.

But since coordinates are whole numbers, and rectangle is positioned, I think a, b, c, d are positive integers, with b > a and d > c, assuming the top is higher (less negative) than bottom.

y-coordinate: below x-axis, so negative. Top side is less negative than bottom side, so if bottom is at y = -c, top should be at y = -d with d < c, so that -d > -c, meaning top is higher.

Set:

• Bottom y-coordinate: y_B = y_A = -c (c > 0)

• Top y-coordinate: y_D = y_C = -d (d > 0)

Since top is above bottom, -d > -c, so d < c.

Similarly, left x-coordinate: x_A = x_D = a (a > 0)

Right x-coordinate: x_B = x_C = b (b > a)

Now, for it to be a rectangle, no issue.

Now, the value we compute for each point is y-coordinate divided by x-coordinate.

Since y is negative and x is positive, this quotient will be negative.

We are to find the smallest value, which since they are negative, the smallest (most negative) means the largest magnitude in negative direction, so actually the most negative number.

But since all are negative, the smallest value corresponds to the one with the largest |y|/x, but since y is negative, it's the one with the largest |y|/x or smallest (most negative).

Let me denote the value as k = y / x

For each point:

A: y_A / x_A = (-c) / a = -c/a

B: y_B / x_B = (-c) / b = -c/b

C: y_C / x_C = (-d) / b = -d/b

D: y_D / x_D = (-d) / a = -d/a

Since a, b, c, d are positive, and b > a > 0, c > d > 0 (because d < c as top is higher).

d < c, so c > d.

Now, all these k values are negative.

We need the smallest k, which since negative, means the one with the largest magnitude, i.e., the most negative.

So, we need to compare -c/a, -c/b, -d/b, -d/a.

Since all are negative, the smallest one will be the one with the largest absolute value divided by the denominator.

For example, -5 is smaller than -3, and | -5 | = 5 > | -3 | = 3.

So, we need to find which of |k| is largest, but since k is negative, min k corresponds to max |k|.

|k| for A is c/a, for B is c/b, for C is d/b, for D is d/a.

So, we need to compare c/a, c/b, d/b, d/a and find the maximum.

Because the max |k| will give the min k.

Now, since b > a > 0, we have 1/b < 1/a, so c/b < c/a, and d/b < d/a.

Also, since c > d > 0, c/a > d/a, and c/b > d/b.

Now, let's list the |k| values:

A: c/a

B: c/b

C: d/b

D: d/a

Since b > a, c/b < c/a, and d/b < d/a.

Since c > d, c/a > d/a, and c/b > d/b.

Now, we need to find which is the largest.

d/a is less than c/a, and c/b is less than c/a, and d/b is less than d/a and also less than c/b since d < c.

d/b < c/b because d < c.

And c/b < c/a because b > a.

So, c/a is larger than c/b, which is larger than d/b, and d/a is larger than d/b but less than c/a.

d/a compared to c/a: since d < c, d/a < c/a.

d/a compared to c/b: not immediately clear.

Similarly, d/a compared to d/b: d/b < d/a since b > a.

And d/a < c/a.

But d/a could be larger or smaller than c/b.

Similarly, we have c/a, c/b, d/b, d/a.

The largest could be either c/a or d/a, but d/a < c/a, so d/a is not larger than c/a.

d/a < c/a as established.

c/b < c/a.

d/b < d/a < c/a.

d/b < d/a, and d/a < c/a, and c/b < c/a.

But is d/a greater than c/b?

Not necessarily.

For example, if a is small, d/a could be large, but c/b might be larger or smaller.

We need to compare d/a and c/b.

d/a vs c/b.

This is (d/a) vs (c/b)

Which is d b vs c a? Since a,b,c,d positive.

d/a > c/b iff d b > c a.

Similarly, d/a < c/b iff d b < c a.

Since b > a and c > d, it could go either way depending on the values.

But in this problem, the coordinates are whole numbers, but we don't have specific values. However, we need to find which point gives the smallest k, and it should be determinable without specific values, or perhaps for all possible such rectangles.

The problem is to find which point, and it might be that for all such rectangles, one particular point always has the smallest k.

Perhaps we need to assume that the rectangle is non-degenerate, so a < b and c > d, all positive integers.

Now, let's consider the differences.

We have four expressions: c/a, c/b, d/b, d/a.

As said, c/b < c/a, d/b < d/a, and d/b < c/b, d/a < c/a.

The largest must be either c/a or d/a, but d/a < c/a, so actually c/a is larger than d/a.

d/a < c/a, yes.

c/a is greater than d/a, and also greater than c/b and d/b.

c/a is greater than c/b because b>a, and greater than d/a because c>d, and d/a < c/a.

d/b is less than d/a < c/a, and less than c/b < c/a.

So, actually, c/a is greater than all others?

Is c/a > d/a? Yes, since c>d.

c/a > c/b? Yes, since b>a.

c/a > d/b? Well, d/b < d/a < c/a, so yes, d/b < c/a.

Similarly, d/a < c/a.

So, for all cases, c/a is the largest among the |k| values.

Therefore, |k| is maximized at A, so k is minimized at A.

k = y/x, for A it's -c/a, and |k| = c/a, which is larger than others, so yes, -c/a is smaller than the others.

For example, suppose a=1, b=2, c=2, d=1.

Then:

A: y/x = (-2)/1 = -2

B: (-2)/2 = -1

C: (-1)/2 = -0.5

D: (-1)/1 = -1

So, min is A: -2.

Another example, suppose a=1, b=3, c=3, d=2.

Then A: -3/1 = -3

B: -3/3 = -1

C: -2/3 ≈ -0.666

D: -2/1 = -2

Min is A: -3.

Another example, suppose d is close to c, but a is small.

Say a=1, b=10, c=10, d=9.

Then A: -10/1 = -10

B: -10/10 = -1

C: -9/10 = -0.9

D: -9/1 = -9

Min is still A: -10.

What if a is larger.

Say a=2, b=3, c=4, d=1.

Then A: -4/2 = -2

B: -4/3 ≈ -1.333

C: -1/3 ≈ -0.333

D: -1/2 = -0.5

Min is A: -2.

Now, is there a case where another point has smaller k?

Suppose we make d large, but d < c.

Say a=1, b=2, c=3, d=2.9, but coordinates are whole numbers, so d and c integers, d < c.

So d <= c-1.

Similarly, b >= a+1.

Minimal case: a=1, b=2, c=2, d=1, as before, A min.

Or a=1, b=2, c=3, d=1.

Then A: -3/1 = -3

B: -3/2 = -1.5

C: -1/2 = -0.5

D: -1/1 = -1

Min A: -3.

Suppose a=3, b=4, c=5, d=4.

A: -5/3 ≈ -1.666

B: -5/4 = -1.25

C: -4/4 = -1

D: -4/3 ≈ -1.333

Min is A: -1.666, since -1.666 < -1.333 < -1.25 < -1.

D is -4/3 ≈ -1.333, which is greater than A's -1.666.

Now, is there a case where D has smaller k than A? But d < c, so |d| < |c|, and same x? No, D has same x as A? A and D both have x=a.

A: (a, -c), D: (a, -d)

Since c > d > 0, |y_A| = c > d = |y_D|, and same x, so |y_A|/x > |y_D|/x, so |k_A| > |k_D|, so k_A < k_D since both negative.

Similarly, compared to B: B has same y as A, but larger x, so |y_B|/x_B = c/b < c/a = |k_A|, since b>a, so |k_B| < |k_A|, so k_B > k_A.

Similarly for C: same x as B, but smaller |y|, since d < c, so |k_C| = d/b < c/b = |k_B| < |k_A|, so |k_C| < |k_A|, so k_C > k_A.

And D: as above, k_D > k_A.

So in all cases, k_A is smaller than k_B, k_C, k_D.

Therefore, point A always gives the smallest value.

But the question is to choose which point, and from this, it should be A.

But let me confirm the points.

In the diagram, A is bottom-left, which is what I assumed.

The problem says "for each of the points we calculate the value of (y-coordinate) ÷ (x-coordinate). Which of the points will give the smallest value?"

And from above, A gives the smallest.

But in the answer, it should be A.

But let me see if there is any trick.

The rectangle is positioned below the x-axis and to the right of y-axis, which I considered.

Coordinates whole numbers, which I used.

And in all my examples, A is smallest.

Is there a case where another point could be smaller? Suppose the rectangle is very flat or very tall.

Suppose it's very tall: a=1, b=2, c=100, d=99.

Then A: -100/1 = -100

B: -100/2 = -50

C: -99/2 = -49.5

D: -99/1 = -99

Still A is smallest.

If it's very flat: a=1, b=100, c=2, d=1.

A: -2/1 = -2

B: -2/100 = -0.02

C: -1/100 = -0.01

D: -1/1 = -1

Min is A: -2.

D is -1, which is larger than -2.

So always A.

But what if a is large, but c is small, but still.

Suppose a=10, b=11, c=2, d=1.

A: -2/10 = -0.2

B: -2/11 ≈ -0.1818

C: -1/11 ≈ -0.0909

D: -1/10 = -0.1

Now, min is D: -0.1? No, -0.2 < -0.1, so A is still smaller.

-0.2 is smaller than -0.1.

Yes.

Is there a case where D is smaller? Only if |y_D|/x_D > |y_A|/x_A, but since same x, and |y_A| > |y_D|, so no.

Similarly, for others.

So, consistently, A gives the smallest value.

Therefore, the answer should be point A.

But the problem says "which of the points", and probably expects the letter.

Now, in the diagram, it's labeled A, B, C, D, with A bottom-left.

So, I think it's A.

But let me see the question again.

"The co-ordinates of the points A,B,C,D are whole numbers. For each of the points we calculate the value of (y-coordinate) ÷ (x-coordinate). Which of the points will give the smallest value?"

And from reasoning, it's A.

Perhaps they want the point, so A.

But in the answer, probably box A.

But let me see if there is any mistake.

One thing: in the beginning, I assumed A is bottom-left, but in the diagram, is it clear?

The diagram shows: y-axis vertical, x-axis horizontal.

Point A is at bottom-left, B bottom-right, C top-right, D top-left.

Yes, as per standard.

And the rectangle is gray, below x-axis, right of y-axis.

So, yes.

Perhaps for some points, if x is zero, but no, since to the right of y-axis, x>0, and whole numbers, so x at least 1.

Similarly, below x-axis, y negative, whole numbers, so y at most -1.

So, no division by zero.

All good.

So, I think it's point A.