Sure! Let's break down each step to help understand how the given numerical result fits the original problem provided in tabular form.
First, we start with the table containing coordinates (a, b) for labels "A," "B," "C," and "D" and their associated counts.
| Label | Coordinates | Count |
|-------|--------------|-------|
| A | (1, 1) | 17 |
| B | (5, 1) | 1 |
| C | (5, 6) | 1 |
| D | (1, 6) | 1 |
### Step-by-Step Solution:
1. Initial Coordinates and Labels:
- The coordinates for label "A" are (1, 1).
- The coordinates for label "B" are (5, 1).
- The coordinates for label "C" are (5, 6).
- The coordinates for label "D" are (1, 6).
2. Transform Coordinates:
- For each label, we need to transform the coordinates using the formula \((a+6, b+1)\).
Let's apply the transformation to each:
- For "A" with coordinates (1, 1):
[tex]\[
(a+6, b+1) = (1+6, 1+1) = (7, 2)
\][/tex]
- For "B" with coordinates (5, 1):
[tex]\[
(a+6, b+1) = (5+6, 1+1) = (11, 2)
\][/tex]
- For "C" with coordinates (5, 6):
[tex]\[
(a+6, b+1) = (5+6, 6+1) = (11, 7)
\][/tex]
- For "D" with coordinates (1, 6):
[tex]\[
(a+6, b+1) = (1+6, 6+1) = (7, 7)
\][/tex]
3. Construct the New Positions Table:
- Using the transformed coordinates, we create a table of new positions:
[tex]\[
\{ "A": (7, 2), "B": (11, 2), "C": (11, 7), "D": (7, 7) \}
\][/tex]
4. Attach Original Counts to New Positions:
- The original counts remain associated with their respective labels:
[tex]\[
\{ "A": 17, "B": 1, "C": 1, "D": 1 \}
\][/tex]
### Original and New Positions with Counts:
By combining our transformed coordinates and retaining original counts, the final tables are:
New Positions with Labels:
[tex]\[
\{
"A": (7, 2),
"B": (11, 2),
"C": (11, 7),
"D": (7, 7)
\}
\][/tex]
New Table with Counts:
[tex]\[
\{
"A": 17,
"B": 1,
"C": 1,
"D": 1
\}
\][/tex]
These results match exactly with the provided numerical results:
[tex]\[
\{
"A": (7, 2),
"B": (11, 2),
"C": (11, 7),
"D": (7, 7)
\},
\{
"A": 17,
"B": 1,
"C": 1,
"D": 1
\}
\][/tex]
So, we have reconstructed the process thoroughly step-by-step to reach the final transformed coordinates and associated counts.
