To find the lengths of the diagonals RT and SU in parallelogram RSTU, we need to equate the expressions associated with each pair of opposite sides from the information provided.
First, let's define the equations given for each diagonal:
- The equation for diagonal RT is \(3m + 2\) on one side and \(4m - 3\) on the opposite side.
- The equation for diagonal SU is \(2n + 2\) on one side and \(8 + n\) on the opposite side.
Since RSTU is a parallelogram, the lengths of the diagonals RT and SU are equal when measured from opposite vertices. Thus, we can set up the following equations based on the property of a parallelogram:
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For diagonal RT: \[ 3m + 2 = 4m - 3 \]
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For diagonal SU: \[ 2n + 2 = 8 + n \]
Let's solve each equation:
1. Solve for m in the RT equation: \[ 3m + 2 = 4m - 3 \] Subtract \(3m\) from both sides: \[ 2 = m - 3 \] Add 3 to both sides: \[ m = 5 \] Now substitute \(m\) back in to find the length of diagonal RT: \[ RT = 3m + 2 = 3(5) + 2 = 15 + 2 = 17 \]
2. Solve for n in the SU equation: \[ 2n + 2 = 8 + n \] Subtract \(n\) from both sides: \[ 2n - n + 2 = 8 \] This simplifies to: \[ n + 2 = 8 \] Subtract 2 from both sides: \[ n = 6 \] Now substitute \(n\) back in to find the length of diagonal SU: \[ SU = 2n + 2 = 2(6) + 2 = 12 + 2 = 14 \]
Combining these results, we find the lengths of the diagonals:
- Diagonal RT is equal to 17.
- Diagonal SU is equal to 14.
Thus, the final answer is:
- Diagonal RT is equal to 17 and diagonal SU is equal to 14.
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