Výplet a rukojeť patří k badmintonové raketě.

They are in the nominative case, because together they form the subject of the sentence:

• výplet = the stringing / strings
• rukojeť = the handle / grip

So the structure is basically:

• Výplet a rukojeť = The strings and the handle
• patří = belong / are part of
• k badmintonové raketě = to a badminton racket / with a badminton racket

This is a very natural question. In Czech, the verb patřit has the same form in the 3rd person singular and 3rd person plural in the present tense:

• on/ona/ono patří = he/she/it belongs
• oni/ony patří = they belong

So in this sentence, patří can mean either belongs or belong, depending on the subject. Since the subject is výplet a rukojeť (two things), here it means belong.

Because the verb patřit is commonly used with the preposition k, and k requires the dative case.

So:

• patřit k + dative = to belong to / to be part of

That is why badmintonová raketa changes to badmintonové raketě:

• nominative: badmintonová raketa
• dative: k badmintonové raketě

This is one of those patterns you should learn together:

• patřit k něčemu = to belong to something

Because in Czech, an adjective must agree with the noun it describes in:

• gender
• number
• case

Here the noun is:

• raketa = feminine singular

After k, it has to be in the dative singular, so both words change:

• badmintonová raketa → badmintonové raketě

So:

• badmintonová → feminine singular nominative
• badmintonové → feminine singular dative
• raketa → nominative
• raketě → dative

Here patřit means something like:

• to belong to
• to be part of
• to go with

In this sentence, it does not mainly mean ownership in the sense of This racket belongs to me. Instead, it means that these parts are associated with or belong as components of a badminton racket.

So a natural English understanding is:

• The stringing and the handle are parts of a badminton racket.
• The strings and the handle belong to a badminton racket.

Výplet means the stringing or string bed of the racket, not the whole racket.

Depending on context, English translations could be:

• stringing
• strings
• string bed

So in this sentence, výplet refers to the woven strings in the racket head.

Because Czech grammatical gender is not determined only by meaning, and not all feminine nouns end in -a.

Rukojeť is a feminine noun of a softer declension type. Its nominative singular form ends in a consonant-like sound, but it is still feminine.

You can see that from agreement patterns in other sentences, for example:

• Ta rukojeť je nová. = That handle is new.

The adjective nová shows that rukojeť is feminine.

A learner-friendly pronunciation guide would be something like:

• rukojeť ≈ roo-ko-yetch

A few helpful details:

• j in Czech sounds like English y in yes
• ě often affects the preceding consonant
• ť is a soft t sound, somewhat like ty said quickly

So the end -jeť sounds softer than a plain English tet or get.

Roughly:

• výplet ≈ VEE-plet

Notes:

• ý is a long ee sound
• stress in Czech is usually on the first syllable, so VÝ-plet

So the first syllable is stressed, and the vowel is long.

No, Czech word order is fairly flexible. This sentence could also be rearranged, for example:

• K badmintonové raketě patří výplet a rukojeť.

That version puts more emphasis on badmintonové raketě.

The original sentence:

• Výplet a rukojeť patří k badmintonové raketě.

is very neutral and straightforward.

Even though word order can change, the case endings still show the grammatical relationships.

Yes, that would also be a correct and natural Czech sentence.

It means:

• The stringing and the handle are parts of a badminton racket.

Compare:

• patří k badmintonové raketě = belong to / go with a badminton racket
• jsou součástí badmintonové rakety = are parts/components of a badminton racket

Both are correct, but součástí is a bit more explicitly about being a component.

Because Czech has no articles.

English uses:

• the badminton racket
• a badminton racket

Czech simply says:

• badmintonová raketa
• badmintonové raketě (in a different case here)

Whether it means a or the depends on context. So this sentence could be understood as:

• the stringing and the handle belong to a badminton racket
• or the stringing and the handle belong to the badminton racket

Context decides.

The dictionary forms are:

• výplet = stringing / strings
• rukojeť = handle
• patřit = to belong
• k = to, toward, with, belonging with (takes dative)
• badmintonový = badminton-related / badminton
• raketa = racket

So the sentence uses:

• patří = present tense of patřit
• badmintonové raketě = dative singular of badmintonová raketa

Yes, grammatically that is a good literal breakdown:

• k = to / toward / with
• badmintonové raketě = badminton racket in the dative singular

But in natural English, with patřit k, you usually translate the whole phrase together:

• patřit k badmintonové raketě = to belong to a badminton racket / to be part of a badminton racket

So it is better to learn the combination as a unit rather than translate k word-for-word every time.