Question

use the image to answer the question.
if quadrilateral ( rstu ) is a parallelogram, what are the lengths of the diagonals of the figure?
(2 points)
diagonal ( rt ) is equal to ( square ) and diagonal ( su ) is equal to ( square ).

Explanation:

Step1: Use the property of parallelogram diagonals bisecting each other

In a parallelogram, the diagonals bisect each other. So, \(3m + 2=4m - 3\) and \(8 + n=2n + 2\).

Solve for \(m\):

\(3m+2 = 4m - 3\)
Subtract \(3m\) from both sides: \(2=m - 3\)
Add \(3\) to both sides: \(m=5\)

Solve for \(n\):

\(8 + n=2n + 2\)
Subtract \(n\) from both sides: \(8=n + 2\)
Subtract \(2\) from both sides: \(n = 6\)

Step2: Calculate the length of diagonal \(RT\)

\(RT=(3m + 2)+(4m - 3)\)
Substitute \(m = 5\):
\(RT=(3\times5 + 2)+(4\times5 - 3)\)
\(=(15 + 2)+(20 - 3)\)
\(=17+17\)
\(RT = 34\)

Step3: Calculate the length of diagonal \(SU\)

\(SU=(8 + n)+(2n + 2)\)
Substitute \(n = 6\):
\(SU=(8 + 6)+(2\times6 + 2)\)
\(=14+(12 + 2)\)
\(=14 + 14\)
\(SU=28\)

Answer:

Diagonal \(RT\) is equal to \(34\) and diagonal \(SU\) is equal to \(28\).