Question

identifying a side length
what is the value of r?
what is the length of segment ef?
units
4r + 19
9r - 6
8r + 3

Explanation:

Step1: Identify equal sides (parallelogram)

In a parallelogram, opposite sides are equal. So, \( 9r - 6 = 8r + 3 \) (wait, no—wait, the markings: the sides with two arrows and the side with \( 4r + 19 \) and the side with one arrow? Wait, no, looking at the diagram: sides \( CF \) (marked \( 9r - 6 \)) and \( DE \) (marked \( 8r + 3 \)) have two arrows, so they are equal. Wait, no, maybe \( CF \) and \( DE \) are equal, and \( CD \) and \( EF \) are equal. Wait, let's re-express:

Wait, the side \( CF \) is \( 9r - 6 \), and side \( DE \) is \( 8r + 3 \). Since they have the same double arrow, they are equal. So:

\( 9r - 6 = 8r + 3 \)

Step2: Solve for r

Subtract \( 8r \) from both sides:

\( 9r - 8r - 6 = 8r - 8r + 3 \)

\( r - 6 = 3 \)

Add 6 to both sides:

\( r - 6 + 6 = 3 + 6 \)

\( r = 9 \)

Step3: Find length of EF (which equals CD, since CD is \( 4r + 19 \) and EF has the same single arrow as CD)

Now, substitute \( r = 9 \) into \( 4r + 19 \):

\( 4(9) + 19 = 36 + 19 = 55 \)

Wait, but also check the other side? Wait, no—wait, EF should equal CD. Let's confirm:

Wait, CD is \( 4r + 19 \), EF is equal to CD (since they have the same single arrow). So with \( r = 9 \), CD is \( 4*9 + 19 = 36 + 19 = 55 \), so EF is 55.

Wait, but let's check the other pair: \( 9r - 6 \) when \( r = 9 \) is \( 81 - 6 = 75 \), and \( 8r + 3 = 72 + 3 = 75 \), so that's correct. So r is 9, EF is 55.

Answer:

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