Question

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if quadrilateral ( rstu ) is a parallelogram, what are the lengths of the diagonals of the figure?

(1 point)

diagonal ( rt ) is equal to ( square ) and diagonal ( su ) is equal to ( square ).

Explanation:

Step1: Use the property of parallelogram diagonals

In a parallelogram, the diagonals bisect each other. So \( RV=VT \) and \( SV = VU \).
For \( RV = VT \), we have \( 3m + 2=4m - 3 \).

Step2: Solve the equation for \( m \)

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

Step3: Use \( SV = VU \) to find \( n \)

Since \( SV = VU \), \( 2n+2=8 + n \).
Subtract \( n \) from both sides: \( 2n+2 - n=8 + n - n \), so \( n+2=8 \).
Subtract 2 from both sides: \( n=8 - 2 = 6 \).

Step4: Calculate the length of diagonal \( RT \)

\( RT=RV + VT=(3m + 2)+(4m - 3) \). Substitute \( m = 5 \):
\( RT=(3\times5 + 2)+(4\times5 - 3)=(15 + 2)+(20 - 3)=17+17 = 28 \).

Step5: Calculate the length of diagonal \( SU \)

\( SU=SV + VU=(2n + 2)+(8 + n) \). Substitute \( n = 6 \):
\( SU=(2\times6+2)+(8 + 6)=(12 + 2)+14=14 + 14=20 \).

Answer:

Diagonal \( RT = 28 \), diagonal \( SU = 20 \)