To prove that there will be a triangle with sides $\sqrt{a}$, $\sqrt{b}$, and $\sqrt{c}$ if $a$, $b$, and $c$ are the sides of a triangle, we can rely on the triangle inequality theorem.

Step 1: Triangle Inequality Theorem for $a$, $b$, and $c$

Since $a$, $b$, and $c$ form a triangle, they must satisfy the triangle inequalities:

1. $a + b > c$
2. $a + c > b$
3. $b + c > a$

Step 2: Transforming to $\sqrt{a}$, $\sqrt{b}$, and $\sqrt{c}$

To ensure that $\sqrt{a}$, $\sqrt{b}$, and $\sqrt{c}$ can also form a triangle, we need to check if they satisfy the triangle inequalities as well:

1. $\sqrt{a} + \sqrt{b} > \sqrt{c}$
2. $\sqrt{a} + \sqrt{c} > \sqrt{b}$
3. $\sqrt{b} + \sqrt{c} > \sqrt{a}$

Step 3: Squaring Both Sides of Each Inequality

To analyze whether these inequalities hold, let’s square both sides of each inequality (since all sides are positive, squaring preserves the inequalities):

1. $(\sqrt{a} + \sqrt{b})^2 > (\sqrt{c})^2$

   • Expanding the left side, we get: $a + b + 2\sqrt{ab} > c$
   • Since $a + b > c$ by the initial triangle inequality, and $2\sqrt{ab} > 0$, this inequality holds true.

2. $(\sqrt{a} + \sqrt{c})^2 > (\sqrt{b})^2$

   • Expanding the left side, we get: $a + c + 2\sqrt{ac} > b$
   • Similarly, $a + c > b$ by the initial triangle inequality, and $2\sqrt{ac} > 0$, so this inequality holds.

3. $(\sqrt{b} + \sqrt{c})^2 > (\sqrt{a})^2$

   • Expanding the left side, we get: $b + c + 2\sqrt{bc} > a$
   • Here as well, $b + c > a$ by the initial triangle inequality, and $2\sqrt{bc} > 0$, so this inequality holds.

Conclusion

Since $\sqrt{a}$, $\sqrt{b}$, and $\sqrt{c}$ satisfy the triangle inequalities, they can form a triangle.

Would you like additional details or explanations on any part of the solution?

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Here are some related questions that might extend this topic:

1. How does the triangle inequality change for different geometric shapes?
2. Can you apply similar principles with other types of functions, like cube roots?
3. What other transformations on side lengths preserve the triangle inequality?
4. How does this proof relate to the concept of similarity in triangles?
5. Are there other geometric properties or transformations that apply to square roots of side lengths?

Tip: When proving inequalities involving square roots, squaring both sides is often useful to simplify the expression—just ensure both sides are positive before doing so.