Lægen undersøgte hendes håndled og sendte hende til røntgen med det samme.

Lægen means the doctor. In Danish, the definite article is usually added to the end of the noun:

• en læge = a doctor
• lægen = the doctor

So lægen undersøgte ... is literally the doctor examined ...

Undersøgte is the past tense of at undersøge = to examine or to investigate.

• jeg undersøger = I examine / am examining
• jeg undersøgte = I examined

In this sentence, the action happened in the past, so Danish uses undersøgte.

When Danish uses a possessive word like min, din, hans, hendes, vores, you normally do not add a separate definite ending to the noun.

So:

• hendes håndled = her wrist
• not hendes håndledet

This works much like English: we say her wrist, not her the wrist.

This is a very common learner question.

Danish uses sin/sit/sine when the possessor is the subject of the same clause.

Here, the subject is lægen = the doctor. But hendes håndled refers to her wrist — the patient's wrist, not the doctor's.

So:

• Lægen undersøgte hendes håndled = The doctor examined her wrist.
• Lægen undersøgte sit håndled would mean The doctor examined his/her own wrist.

Because the wrist belongs to someone other than the subject, Danish uses hendes.

Because hende is the object form of she.

• hun = she
• hende = her

After a verb like sendte = sent, you need the object form:

• Lægen sendte hende ... = The doctor sent her ...

Just like in English, you say sent her, not sent she.

Håndled means wrist.

It is a compound noun:

• hånd = hand
• led = joint

So it literally means something like hand joint.

Til røntgen means something like for an X-ray or to X-ray in a hospital/medical context.

Røntgen can refer to:

• X-rays as a technique
• the X-ray department
• an X-ray examination

So sendte hende til røntgen does not mean he physically mailed her somewhere. It means the doctor referred/sent her for an X-ray examination.

In medical and institutional expressions, Danish often leaves out an article where English might use one.

So:

• til røntgen = for X-ray / to the X-ray department / for an X-ray

This is similar to other expressions where the place or function matters more than a full noun phrase.

The sentence has two coordinated verbs joined by og = and:

• Lægen undersøgte hendes håndled
• og sendte hende til røntgen med det samme

The subject lægen applies to both verbs, so Danish does not need to repeat it.

In English, we do the same:

• The doctor examined her wrist and sent her for an X-ray.

We do not have to say The doctor examined her wrist and the doctor sent her ...

Med det samme means immediately, right away, or at once.

Literally, it is something like with the same, but you should learn it as a fixed expression.

So:

• sendte hende til røntgen med det samme = sent her for an X-ray immediately

A close synonym is straks.

That is a very natural place for a time adverbial in Danish.

The core clause is:

• Lægen sendte hende til røntgen

Then med det samme is added to say when/how quickly:

• Lægen sendte hende til røntgen med det samme

Danish word order is often flexible with adverbials, but placing med det samme at the end sounds very normal here.

It is literally sent, but in this context English often translates it more naturally as referred or sent ... for an X-ray.

So depending on style, you might understand it as:

• The doctor sent her for an X-ray.
• The doctor referred her for an X-ray.

All of these match the Danish meaning in context.