The image shows a triangle $ABC$ where $AB = AC$, implying it's an isosceles triangle. The segment $AD$ inside the triangle forms two smaller triangles, and a length of $\sqrt{2}$ is indicated for the segment $BD$.

Let me break down the steps to solve for $BC$.

Key points:

• $AB = AC$ indicates an isosceles triangle.
• The lengths labeled are:
  • $AD = 1$
  • $CD = 2$
  • $BD = \sqrt{2}$

From here, we will use the properties of triangles and the Pythagorean theorem to calculate $BC$. Let me perform the calculations.The solution yields the following values:

• The length of $AB = AC$ is approximately $1.73$.
• The length of $BC$ is approximately $3.41$.

These results were derived using the Pythagorean theorem and the properties of the isosceles triangle $ABC$.

Would you like further details or have any other questions?

Here are 5 related questions for further exploration:

1. How do we apply the Pythagorean theorem in more complex geometric problems?
2. Can we calculate the area of triangle $ABC$ using these lengths?
3. What is the significance of an isosceles triangle in geometry?
4. How can we confirm the accuracy of the given length $\sqrt{2}$ in this context?
5. How would the solution change if triangle $ABC$ were scalene instead of isosceles?

Tip: In any triangle, when solving for unknown sides, breaking it into smaller right triangles often simplifies the problem significantly.