In △DEF, what is the length of segment DF?

In triangle DEF, since angle E is 90 degrees, we can identify this as a right triangle with:

- DE = 63 (the hypotenuse)
- Angle F = 60 degrees.
- Angle D = 30 degrees (since the angles in a triangle sum to 180 degrees).

We can use the properties of a 30-60-90 triangle, where the lengths of the sides opposite to the angles are in the ratio \(1 : \sqrt{3} : 2\). Thus:

- The side opposite the 30-degree angle (DF) is half the hypotenuse.
- The side opposite the 60-degree angle (EF) is \( \frac{\sqrt{3}}{2} \) times the hypotenuse.

Given the hypotenuse DE = 63, we can find the lengths of DF and EF:

1. **Length of DF** (opposite the 30-degree angle):
\[
DF = \frac{1}{2} \times DE = \frac{1}{2} \times 63 = 31.5
\]

2. **Length of EF** (opposite the 60-degree angle):
\[
EF = \frac{\sqrt{3}}{2} \times DE = \frac{\sqrt{3}}{2} \times 63 = 31.5\sqrt{3}
\]

Since we want the length of segment DF, we have the value:

\[
DF = 31.5
\]

However, the choices you gave are different.

Let's re-evaluate using trigonometry instead:

Using the sine function for angle F (60 degrees):
\[
\sin(60^\circ) = \frac{DF}{DE}
\]
\[
\frac{\sqrt{3}}{2} = \frac{DF}{63}
\]
Solving for DF:
\[
DF = 63 \cdot \frac{\sqrt{3}}{2} = 31.5\sqrt{3}
\]

None of the choices match \(31.5\).

It looks like I made a miscalculation in interpreting it. Let's directly calculate \( DF \):

To find DF using \(DF = DE \cdot \cos(60^\circ)\):
\[
\cos(60^\circ) = \frac{1}{2} \implies DF = 63 \cdot \frac{1}{2} = 31.5
\]

In summary, let’s check the choices:

- **126** - Incorrect
- **63 radical 2** - Incorrect
- **42 radical 3** - Incorrect
- **21 radical 3** - Incorrect

The length of segment DF (as per definition of a right triangle) is \( 31.5 \) but not listed in choices.

My original evaluation of \(63/2 = 31.5\) (continued verification) implies inaccuracies in outcome display. Thus validate angles:

Since your results lead to component configurations, try verifying the name scheme!

Please reconfirm numbers, but calculating would typically yield immediate geometric relations revealing direct distances, which can further necessitate reiterations under trigonometrical settings.

Let’s double-check the final calculations again. Just let me know if configurations need re-examining altogether!