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 \( n \)

First, we need to find the distance \( |KL| \) between points \( K(-1, 3) \) and \( L(2, 3) \). The formula for distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Applying the coordinates of \( K \) and \( L \): \[ |KL| = \sqrt{(2 - (-1))^2 + (3 - 3)^2} = \sqrt{(2 + 1)^2 + 0} = \sqrt{3^2} = 3 \] Now we are given that \( |KL| = |RQ| \). Next, we need to find \( |RQ| \) between points \( R(m-1, n-2) \) and \( Q(6, -3) \): \[ |RQ| = \sqrt{(6 - (m-1))^2 + (-3 - (n-2))^2} \] This can be simplified to: \[ |RQ| = \sqrt{(6 - m + 1)^2 + (-3 - n + 2)^2} = \sqrt{(7 - m)^2 + (-1 - n)^2} \] Setting this equal to the distance |KL|: \[ \sqrt{(7 - m)^2 + (-1 - n)^2} = 3 \] Squaring both sides gives: \[ (7 - m)^2 + (-1 - n)^2 = 9 \] Now we expand this equation: \[ (7 - m)^2 + (n + 1)^2 = 9 \] Let’s solve this equation step by step. First, we can set \( (7 - m) = a \) and \( (n + 1) = b \), leading to: \[ a^2 + b^2 = 9 \] This represents a circle with radius \( 3 \) centered at the origin in terms of \( a \) and \( b \). We can find specific integer solutions. For simplicity, let’s evaluate cases where \( a \) and \( b \) can yield integer squares. 1. If \( a = 0 \), then \( b^2 = 9 \Rightarrow b = 3 \) or \( b = -3 \) - \( 7 - m = 0 \Rightarrow m = 7 \) and \( n + 1 = 3 \Rightarrow n = 2 \) → solution: \( (m, n) = (7, 2) \) - \( 7 - m = 0 \Rightarrow m = 7 \) and \( n + 1 = -3 \Rightarrow n = -4 \) → solution: \( (m, n) = (7, -5) \) 2. If \( a = 3 \), then \( b^2 = 0 \Rightarrow b = 0 \) - \( 7 - m = 3 \Rightarrow m = 4 \) and \( n + 1 = 0 \Rightarrow n = -1 \) → solution: \( (m, n) = (4, -1) \) 3. If \( a = -3 \), then \( b^2 = 0 \Rightarrow b = 0 \) - \( 7 - m = -3 \Rightarrow m = 10 \) and \( n + 1 = 0 \Rightarrow n = -1 \) → solution: \( (m, n) = (10, -1) \) 4. If \( a = 2 \) or \( a = -2 \), we can find similar assignments giving other integer pairs based on the squares. Verifying these: We can take pairs [\( (m, n) (7, 2) ,( 4, -1), (10, -1) \)]. Lastly, one valid solution for \( (m, n) = (4, -1) \) is commonly accepted after ensuring all conditions are intact. Thus, the value of \( m \) and \( n \) could be: \[ m = 4, \; n = -1 \] or others depending on the integer conditions. Final solutions, you can pick your pair!