To determine which value lies between two points on a number line, we need to evaluate the expressions provided in the answer choices and see which one falls between two specific values.

Let's evaluate each option step-by-step:

Option A: $\sqrt{2} + 2$

• $\sqrt{2} \approx 1.414$
• So, $\sqrt{2} + 2 \approx 1.414 + 2 = 3.414$

Option B: $\frac{5}{\sqrt{3}}$

• $\sqrt{3} \approx 1.732$
• So, $\frac{5}{\sqrt{3}} \approx \frac{5}{1.732} \approx 2.887$

Option C: $\frac{19}{5\sqrt{21}}$

• $\sqrt{21} \approx 4.583$
• So, $5\sqrt{21} \approx 5 \times 4.583 = 22.915$
• $\frac{19}{5\sqrt{21}} \approx \frac{19}{22.915} \approx 0.829$

Option D: $\sqrt{5} - 1$

• $\sqrt{5} \approx 2.236$
• So, $\sqrt{5} - 1 \approx 2.236 - 1 = 1.236$

Summary of Evaluations

• Option A: $\sqrt{2} + 2 \approx 3.414$
• Option B: $\frac{5}{\sqrt{3}} \approx 2.887$
• Option C: $\frac{19}{5\sqrt{21}} \approx 0.829$
• Option D: $\sqrt{5} - 1 \approx 1.236$

Conclusion

To find which value lies between the two others on a number line, we should order these values from smallest to largest:

1. Option C: $\frac{19}{5\sqrt{21}} \approx 0.829$
2. Option D: $\sqrt{5} - 1 \approx 1.236$
3. Option B: $\frac{5}{\sqrt{3}} \approx 2.887$
4. Option A: $\sqrt{2} + 2 \approx 3.414$

The value between two other values is Option D: $\sqrt{5} - 1$.

Would you like more details on any of these calculations or have any other questions?

Here are 5 related questions for further exploration:

1. How do we rationalize the denominator in expressions like Option B?
2. How can we approximate square roots without a calculator?
3. Why is it important to compare numerical values on a number line?
4. What are the properties of square roots that help simplify expressions?
5. How do different operations (addition, subtraction, division) affect the placement of numbers on a number line?

Tip: When approximating square roots, remember that they are non-repeating and non-terminating decimals, making rough estimations helpful for quick calculations.