To find the lengths of the diagonals of the parallelogram RSTU (with vertices at coordinates), we need to apply the properties of a parallelogram and potentially the distance formula.
Given the points R, S, T, and U are specified by their coordinates:
- \( R = (3m + 2, 4m - 3) \)
- \( S = (V, 8 + n) \)
- \( T = (3m + 2, 4m - 3) \) (assuming some coordinates here)
- \( U = (8 + n, 2n + 2) \)
In a parallelogram, the diagonals bisect each other, which means their midpoints will be the same.
To find the lengths of the diagonals RT and SU, we will use the distance formula:
For diagonal RT:
- Let \( R(m, n) = (3m+2, 4m-3) \)
- Let \( T(m', n') = (8+n, 4m-3) \)
The distance \( RT \) is given by the distance formula:
\[ RT = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
For diagonal SU:
- Let \( S(x_1, y_1) = (V, 8+n) \)
- Let \( U(x_2, y_2) = (8+n, 2n+2) \)
Similarly, the distance \( SU \) can be calculated using the same distance formula.
The outcome will depend on the particular values of \( m \) and \( n \) that you might be referring to.
If there are no specific coordinates available for S and U, we can substitute them with their general coordinates given.
Now, to solve:
Diagonal RT: \( RT = \sqrt{(x_T - x_R)^2 + (y_T - y_R)^2} \) Using the coordinates given (assuming no errors in coordinates input):
- Let’s assume the computations are based on values of \(m\) and \(n\).
Diagonal SU: \( SU = \sqrt{(x_U - x_S)^2 + (y_U - y_S)^2} \)
Complete evaluating these two distance formulas to find lengths \( RT \) and \( SU \).
After performing algebraic simplification you should arrive at specific lengths stated with the variables \( m \) and \( n \).
Conclusion: For specific numerical results of diagonals \( RT \) and \( SU \), it is essential first to plug in the exact values for \( m \) and \( n \) based on your further details to get the length where you mentioned the first blank would equal to symbolically solve for \( RT \) and \( SU \).
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