In this problem, Triangle $C$ is a scaled copy of Triangle $B$. This means that the triangles are similar, so their corresponding angles are equal, and their sides are proportional.

Given:

• In Triangle $B$:
  • Angles: $70^\circ, 70^\circ, 40^\circ$
  • Side lengths: $24, 24$
• In Triangle $C$:
  • Angles are identical to those in Triangle $B$: $70^\circ, 70^\circ, 40^\circ$
  • The side corresponding to 24 in Triangle $B$ is 30 in Triangle $C$.

Since the triangles are similar, we can set up a proportion to find the scale factor from Triangle $B$ to Triangle $C$.

The scale factor $k$ can be calculated as: $k = \frac{\text{Side in Triangle C}}{\text{Corresponding side in Triangle B}} = \frac{30}{24} = 1.25$

To find the unknown side $a$ in Triangle $C$, which corresponds to the 24 units side in Triangle $B$, we apply the scale factor: $a = 24 \times 1.25 = 30$

Answer: The value of $a$ is $30$.

Would you like further details on this solution or have any additional questions?

Here are some related questions:

1. How can we verify if two triangles are similar?
2. What are the criteria for similarity in triangles?
3. How do you calculate the scale factor between similar shapes?
4. What is the effect of scaling on angles in similar triangles?
5. How can you find unknown side lengths in similar polygons?

Tip: When working with similar figures, remember that corresponding angles remain equal while corresponding sides are proportional.