This question is about finding specific lengths in a triangle where points $D$, $E$, and $F$ are midpoints of the sides of triangle $\triangle ABC$. Given the information:

• $DF = 34$
• $AC = 48$
• $BC = 78$

We are tasked with finding:

1. $DE$
2. $AB$
3. $BE$

Since $D$, $E$, and $F$ are midpoints, the triangle formed by connecting the midpoints is governed by the Midsegment Theorem, which states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

Step 1: Finding $DE$

Using the Midsegment Theorem:

• $DE$ is a midsegment, parallel to $AB$.
• $DE = \frac{1}{2} AB$.

However, we do not yet know $AB$, so we'll solve for $AB$ next.

Step 2: Finding $AB$

To find $AB$, we can use the fact that in quadrilateral $ADFE$, $DF$ is parallel to $AB$ by the Midsegment Theorem. Since $DF = 34$, and $DF$ is half of $AB$:

• $AB = 2 \times DF = 2 \times 34 = 68$.

Step 3: Finding $DE$

Now that we know $AB = 68$, we can find $DE$:

• $DE = \frac{1}{2} AB = \frac{1}{2} \times 68 = 34$.

Step 4: Finding $BE$

Finally, $BE$ is half of $BC$ because $E$ is the midpoint of $BC$. Therefore:

• $BE = \frac{1}{2} BC = \frac{1}{2} \times 78 = 39$.

Summary of Results:

1. $DE = 34$
2. $AB = 68$
3. $BE = 39$

Would you like further details or explanations? Here are five related questions:

1. How is the Midsegment Theorem used in other geometric shapes?
2. What properties are shared by midsegments in different types of triangles?
3. How can the Pythagorean theorem be used in conjunction with the Midsegment Theorem?
4. Can the lengths found here help determine the area of triangle $\triangle ABC$?
5. What are the relationships between other midsegments in triangle geometry?

Tip: The Midsegment Theorem is powerful for simplifying complex triangle problems and can also help in coordinate geometry!