The vertices of a quadrilateral are $$ \mathrm{K}(-1,3) $$ and 6. $$ L(2,3) $$ and $$ Q(6,-3) $$ and $$ R(m-1, n-2) $$ if $$ |K L|=|R Q| $$ find $$ m $$ and $$ n $$

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To find the values of $$ m $$ and $$ n $$ such that the lengths of segments $$ KL $$ and $$ RQ $$ are equal, we first need to calculate the length of segment $$ KL $$.

The coordinates of points $$ K $$ and $$ L $$ are:
- $$ K(-1, 3) $$
- $$ L(2, 3) $$

The length of segment $$ KL $$ can be calculated using the distance formula:
$$
|KL| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$
Substituting the coordinates of $$ K $$ and $$ L $$:
$$
|KL| = \sqrt{(2 - (-1))^2 + (3 - 3)^2} = \sqrt{(2 + 1)^2 + 0^2} = \sqrt{3^2} = 3
$$

Next, we need to find the length of segment $$ RQ $$, where the coordinates of points $$ R $$ and $$ Q $$ are:
- $$ R(m-1, n-2) $$
- $$ Q(6, -3) $$

Using the distance formula again for segment $$ RQ $$:
$$
|RQ| = \sqrt{(6 - (m-1))^2 + (-3 - (n-2))^2}
$$
This simplifies to:
$$
|RQ| = \sqrt{(6 - m + 1)^2 + (-3 - n + 2)^2} = \sqrt{(7 - m)^2 + (-1 - n)^2}
$$

We set the lengths equal to each other:
$$
|KL| = |RQ| \implies 3 = \sqrt{(7 - m)^2 + (-1 - n)^2}
$$

Squaring both sides to eliminate the square root gives:
$$
9 = (7 - m)^2 + (-1 - n)^2
$$

Expanding the right side:
$$
9 = (7 - m)^2 + (n + 1)^2
$$

Now we can express this as:
$$
(7 - m)^2 + (n + 1)^2 = 9
$$

This equation represents a circle centered at $$ (7, -1) $$ with a radius of 3. We can find integer solutions for $$ m $$ and $$ n $$ that satisfy this equation.

To find integer solutions, we can test values for $$ (7 - m) $$ and $$ (n + 1) $$ that satisfy the equation:
1. $$ (7 - m)^2 + (n + 1)^2 = 9 $$

The possible integer pairs $$ (a, b) $$ such that $$ a^2 + b^2 = 9 $$ are:
- $$ (3, 0) $$
- $$ (0, 3) $$
- $$ (3, 0) $$
- $$ (-3, 0) $$
- $$ (0, -3) $$
- $$ (2, 2) $$
- $$ (2, -2) $$
- $$ (-2, 2) $$
- $$ (-2, -2) $$

Now we can find corresponding values of $$ m $$ and $$ n $$:
1. For $$ (3, 0) $$:
   - $$ 7 - m = 3 $$ → $$ m = 4 $$
   - $$ n + 1 = 0 $$ → $$ n = -1 $$

2. For $$ (0, 3) $$:
   - $$ 7 - m = 0 $$ → $$ m = 7 $$
   - $$ n + 1 = 3 $$ → $$ n = 2 $$

3. For $$ (-3, 0) $$:
   - $$ 7 - m = -3 $$ → $$ m = 10 $$
   - $$ n + 1 = 0 $$ → $$ n = -1 $$

4. For $$ (0, -3) $$:
   - $$ 7 - m = 0 $$ → $$ m = 7 $$
   - $$ n + 1 = -3 $$ → $$ n = -4 $$

5. For $$ (2, 2) $$:
   - $$ 7 - m = 2 $$ → $$ m = 5 $$
   - $$ n + 1 = 2 $$ → $$ n = 1 $$

6. For $$ (2, -2) $$:
   - $$ 7 - m = 2 $$ → $$ m = 5 $$
   - $$ n + 1 = -2 $$ → $$ n = -3 $$

7. For $$ (-2, 2) $$:
   - $$ 7 - m = -2 $$ → $$ m = 9 $$
   - $$ n + 1 = 2 $$ → $$ n = 1 $$

8. For $$ (-2, -2) $$:
   - $$ 7 - m = -2 $$ → $$ m = 9 $$
   - $$ n + 1 = -2 $$ → $$ n = -3 $$

Thus, the integer solutions for $$ (m, n) $$ are:
- $$ (4, -1) $$
- $$ (7, 2) $$
- $$ (10, -1) $$
- $$ (7, -4) $$
- $$ (5, 1) $$
- $$ (5, -3) $$
- $$ (9, 1) $$
- $$ (9, -3) $$

In conclusion, the values of $$ m $$ and $$ n $$ can be:
- $$ m = 4, n = -1 $$
- $$ m = 7, n = 2 $$
- $$ m = 10, n = -1 $$
- $$ m = 7, n = -4 $$
- $$ m = 5, n = 1 $$
- $$ m = 5, n = -3 $$
- $$ m = 9, n = 1 $$
- $$ m = 9, n = -3 $$

You can choose any of these pairs based on the context of the problem.
The values of $$ m $$ and $$ n $$ are: - $$ m = 4, n = -1 $$ - $$ m = 7, n = 2 $$ - $$ m = 10, n = -1 $$ - $$ m = 7, n = -4 $$ - $$ m = 5, n = 1 $$ - $$ m = 5, n = -3 $$ - $$ m = 9, n = 1 $$ - $$ m = 9, n = -3 $$

The vertices of a quadrilateral are \( \mathrm{K}(-1,3) \) and 6. \( L(2,3) \) and \( Q(6,-3) \) and \( R(m-1, n-2) \) if \( |K L|=|R Q| \) find \( m \) and \( n \)

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