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:
$$
|KL| = |RQ| \implies 3 = \sqrt{(7 - m)^2 + (-1 - n)^2}
$$

Next, we square both sides to eliminate the square root:
$$
3^2 = (7 - m)^2 + (-1 - n)^2
$$
$$
9 = (7 - m)^2 + (-1 - n)^2
$$

Now we expand the right side:
$$
(7 - m)^2 + (-1 - n)^2 = (7 - m)^2 + (n + 1)^2
$$
Let’s expand both squares:
$$
(7 - m)^2 = 49 - 14m + m^2
$$
$$
(n + 1)^2 = n^2 + 2n + 1
$$
Combining these, we have:
$$
9 = 49 - 14m + m^2 + n^2 + 2n + 1
$$
$$
9 = m^2 + n^2 - 14m + 2n + 50
$$
Rearranging gives:
$$
m^2 + n^2 - 14m + 2n + 50 - 9 = 0
$$
$$
m^2 + n^2 - 14m + 2n + 41 = 0
$$

This is a quadratic equation in terms of $$ m $$ and $$ n $$. To find specific values for $$ m $$ and $$ n $$, we can try to express $$ n $$ in terms of $$ m $$ or vice versa.

However, we can also analyze the equation geometrically or numerically.

To find integer solutions, we can try some values for $$ m $$ and see if we can find corresponding $$ n $$ values that satisfy the equation.

Let’s try $$ m = 7 $$:
$$
7^2 + n^2 - 14(7) + 2n + 41 = 0
$$
$$
49 + n^2 - 98 + 2n + 41 = 0
$$
$$
n^2 + 2n - 8 = 0
$$
Factoring gives:
$$
(n + 4)(n - 2) = 0
$$
Thus, $$ n = -4 $$ or $$ n = 2 $$.

So we have two pairs:
1. $$ (m, n) = (7, -4) $$
2. $$ (m, n) = (7, 2) $$

We can also try other values for $$ m $$ to find more pairs, but these are valid solutions.

Thus, the values of $$ m $$ and $$ n $$ that satisfy the condition $$ |KL| = |RQ| $$ are:
$$
\boxed{(7, -4)} \quad \text{and} \quad \boxed{(7, 2)}
$$
The values of $$ m $$ and $$ n $$ that satisfy the condition are $$ (7, -4) $$ and $$ (7, 2) $$.

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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