Question

what is the value of r?
what is the length of segment ef?
units

Explanation:

Step1: Identify the figure as a parallelogram

In a parallelogram, opposite sides are equal. So, \( 9r - 6 = 8r + 3 \) (since \( CF \) and \( DE \) are opposite sides) and \( 4r + 19 = EF \) (since \( CD \) and \( EF \) are opposite sides). First, solve for \( r \) using \( 9r - 6 = 8r + 3 \).

Step2: Solve for \( r \)

Subtract \( 8r \) from both sides: \( 9r - 8r - 6 = 8r - 8r + 3 \)
Simplify: \( r - 6 = 3 \)
Add 6 to both sides: \( r - 6 + 6 = 3 + 6 \)
So, \( r = 9 \)

Step3: Find the length of \( EF \)

Now that \( r = 9 \), substitute into \( 4r + 19 \) (since \( EF = CD \) in the parallelogram).
\( EF = 4(9) + 19 \)
Calculate: \( 36 + 19 = 55 \)

Answer:

The value of \( r \) is \( 9 \).
The length of segment \( EF \) is \( 55 \) units.