figure cdef is a parallelogram.
which equations are correct? select three options.
□ n = 10
□ n = 7
□ cf = 59
□ fe = 42
□ cd = 30

Question

Accepted Answer

# Explanation:
## Step1: Recall properties of parallelograms
In a parallelogram, opposite sides are equal. So, $ CF = DE $ and $ CD = FE $, also $ CF = 6n - 1 $, $ DE = 5n + 9 $, $ CD = 4n + 2 $, $ FE = 6n - 1 $? Wait, no, looking at the figure, $ CF $ and $ DE $ are opposite sides, $ CD $ and $ FE $ are opposite sides. So $ CF = DE $ (since $ CF = 6n - 1 $, $ DE = 5n + 9 $) and $ CD = FE $ ( $ CD = 4n + 2 $, $ FE $ should be equal to $ CD $ or $ CF $? Wait, let's correct: In parallelogram $ CDEF $, opposite sides are equal. So $ CD = FE $ and $ CF = DE $. So $ CD = 4n + 2 $, $ FE $ is... Wait, the sides: $ CF = 6n - 1 $, $ DE = 5n + 9 $, so $ CF = DE $ (opposite sides). So set $ 6n - 1 = 5n + 9 $.

## Step2: Solve for $ n $
$ 6n - 1 = 5n + 9 $
Subtract $ 5n $ from both sides: $ 6n - 5n - 1 = 9 $
$ n - 1 = 9 $
Add 1 to both sides: $ n = 10 $? Wait, no: $ 6n - 5n = 9 + 1 $ → $ n = 10 $? Wait, $ 6n -1 =5n +9 $ → $ n = 10 $. Wait, but let's check the other option $ n =7 $. Wait, maybe I mixed up the sides. Wait, maybe $ CD $ and $ FE $ are opposite, and $ CF $ and $ DE $ are opposite. Wait, the labels: $ CDEF $, so vertices are $ C, D, E, F $, so sides: $ CD $, $ DE $, $ EF $, $ FC $. So $ CD \parallel EF $ and $ CD = EF $; $ DE \parallel FC $ and $ DE = FC $. So $ CD = 4n + 2 $, $ EF = $? Wait, the side $ EF $ is labeled as... Wait, the figure has $ CD = 4n + 2 $, $ CF = 6n -1 $, $ DE = 5n +9 $, $ FE $ is... Wait, maybe $ CD = FE $ and $ CF = DE $. So $ CD = 4n + 2 $, $ FE $ should be equal to $ CD $, and $ CF = 6n -1 $, $ DE = 5n +9 $, so $ CF = DE $ → $ 6n -1 = 5n +9 $. Solving: $ 6n -5n = 9 +1 $ → $ n =10 $. Wait, but let's check $ n =7 $: if $ n=7 $, $ 6n -1 =41 $, $ 5n +9=44 $, not equal. If $ n=10 $, $ 6n -1=59 $, $ 5n +9=59 $, so that works. Then $ CD =4n +2 =4*10 +2=42 $? Wait, no, $ CD =4n +2 $, if $ n=10 $, $ CD=42 $, but $ FE $ should be equal to $ CD $, so $ FE=42 $. Wait, but the options: $ FE=42 $ is an option. Also, $ CF=6n -1=59 $ (since $ n=10 $, $ 6*10 -1=59 $), so $ CF=59 $ is an option. Also, $ n=10 $ is an option. Wait, but let's check $ n=7 $: $ 6*7 -1=41 $, $ 5*7 +9=44 $, not equal. So $ n=10 $ is correct. Then $ CD=4n +2=4*10 +2=42 $? Wait, no, the option $ CD=30 $: if $ n=7 $, $ CD=4*7 +2=30 $, so $ CD=30 $ when $ n=7 $. Wait, maybe I mixed up the opposite sides. Let's re-express: Maybe $ CD $ and $ FE $ are opposite, and $ CF $ and $ DE $ are opposite. Wait, maybe the sides: $ CD = 4n +2 $, $ FE = 6n -1 $ (opposite), and $ CF = 6n -1 $? No, the labels: $ CF $ is from $ C $ to $ F $, $ DE $ is from $ D $ to $ E $, so $ CF \parallel DE $ and $ CF = DE $. $ CD $ is from $ C $ to $ D $, $ FE $ is from $ F $ to $ E $, so $ CD \parallel FE $ and $ CD = FE $. So $ CD = FE $ → $ 4n + 2 = 6n -1 $ (if $ FE = 6n -1 $)? Wait, that would be another equation. Let's check both possibilities.

Case 1: $ CF = DE $ ( $ CF = 6n -1 $, $ DE = 5n +9 $ )
$ 6n -1 = 5n +9 $ → $ n=10 $
Then $ CD =4n +2=42 $, $ FE = CD=42 $ (since $ CD=FE $), $ CF=6n -1=59 $, $ DE=5n +9=59 $. So options: $ n=10 $, $ FE=42 $, $ CF=59 $.

Case 2: $ CD = FE $ ( $ CD=4n +2 $, $ FE=6n -1 $ )
$ 4n +2 =6n -1 $ → $ -2n = -3 $ → $ n=1.5 $, which is not an option (options are $ n=10 $, $ n=7 $). So Case 1 is correct.

Now check $ n=7 $: If $ n=7 $, $ CD=4*7 +2=30 $, then $ FE $ should be equal to $ CD $ (if $ CD=FE $), so $ FE=30 $, but $ FE $ is labeled as... Wait, maybe the opposite sides are $ CD $ and $ FE $, with $ FE=6n -1 $, so $ 4n +2 =6n -1 $ → $ n=1.5 $, not integer. So the correct equation is $ CF=DE $ ( $ 6n -1=5n +9 $ ) → $ n=10 $. Then:

- $ n=10 $: correct.
- $ FE=CD=4n +2=42 $: correct (since $ CD=FE $ in parallelogram).
- $ CF=6n -1=59 $: correct (since $ CF=DE=5n +9=59 $ when $ n=10 $).
- $ CD=30 $: when $ n=7 $, $ CD=4*7 +2=30 $, but $ n=7 $ would mean $ CF=6*7 -1=41 $, $ DE=5*7 +9=44 $, which are not equal, so $ n=7 $ is incorrect.
- $ CD=30 $ is correct only if $ n=7 $, but $ n=7 $ is incorrect, so $ CD=30 $ is incorrect? Wait, no, if $ n=7 $, $ CD=30 $, but does $ n=7 $ satisfy the parallelogram? Let's check: if $ n=7 $, $ CF=6*7 -1=41 $, $ DE=5*7 +9=44 $, 41≠44, so not a parallelogram. So $ n=7 $ is wrong. So correct options: $ n=10 $, $ FE=42 $, $ CF=59 $. Wait, but the options include $ CD=30 $, which is when $ n=7 $, but $ n=7 $ is wrong. Wait, maybe I made a mistake in identifying opposite sides. Let's look at the figure again: the sides are $ CF=6n -1 $, $ CD=4n +2 $, $ DE=5n +9 $, $ FE=6n +9 $? Wait, no, the user's figure: $ FE $ is labeled $ 5n +9 $? Wait, no, the original figure: $ CF=6n -1 $, $ CD=4n +2 $, $ DE=5n +9 $, $ FE=6n +9 $? No, the user's image: $ FE $ is labeled $ 5n +9 $? Wait, no, the text: "5n +9" is next to $ DE $? Wait, the user's image: the sides: $ CF $ is $ 6n -1 $, $ CD $ is $ 4n +2 $, $ DE $ is $ 5n +9 $, $ FE $ is $ 6n +9 $? No, the user wrote: "5n +9" next to $ E $ and $ D $, so $ DE=5n +9 $, $ FE $ is... Wait, maybe the opposite sides are $ CF $ and $ DE $ (so $ CF=DE $), and $ CD $ and $ FE $ (so $ CD=FE $). So $ CF=6n -1 $, $ DE=5n +9 $, so $ 6n -1=5n +9 $ → $ n=10 $. Then $ CD=4n +2=42 $, $ FE=CD=42 $ (so $ FE=42 $ is correct). $ CF=6n -1=59 $ (correct). $ n=10 $ (correct). Now, if we take $ n=7 $, $ CD=4*7 +2=30 $, then $ FE $ should be $ CD=30 $, but $ FE $ is $ 6n -1=41 $ (if $ FE=CF $? No, that's not opposite. So $ CD=30 $ is correct only if $ n=7 $, but $ n=7 $ is wrong, so $ CD=30 $ is incorrect. So the correct options are $ n=10 $, $ FE=42 $, $ CF=59 $. Wait, but let's check the options:

Options:
- $ n=10 $: correct (from $ 6n -1=5n +9 $)
- $ n=7 $: incorrect (as shown)
- $ CF=59 $: correct ( $ 6*10 -1=59 $)
- $ FE=42 $: correct ( $ CD=4*10 +2=42 $, $ FE=CD $)
- $ CD=30 $: correct only if $ n=7 $, but $ n=7 $ is incorrect, so $ CD=30 $ is incorrect.

Wait, but maybe the opposite sides are $ CD $ and $ FE $ with $ FE=6n -1 $, so $ 4n +2=6n -1 $ → $ n=1.5 $, not integer. So the only way is $ CF=DE $ → $ n=10 $, $ FE=42 $, $ CF=59 $. So the three correct options are $ n=10 $, $ FE=42 $, $ CF=59 $.
# Answer:
n = 10, FE = 42, CF = 59 (the options corresponding to these are the correct ones: n = 10, FE = 42, CF = 59)

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QUESTION IMAGE

Question

Explanation:

Step1: Recall properties of parallelograms

Step2: Solve for \( n \)

Answer: