To find the length of the line segment between two points $C(-2, 1)$ and $D(4, 5)$, we use the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Here:

• $(x_1, y_1) = (-2, 1)$
• $(x_2, y_2) = (4, 5)$

Steps:

1. Subtract the coordinates:

   • $x_2 - x_1 = 4 - (-2) = 4 + 2 = 6$
   • $y_2 - y_1 = 5 - 1 = 4$

2. Square the differences:

   • $(x_2 - x_1)^2 = 6^2 = 36$
   • $(y_2 - y_1)^2 = 4^2 = 16$

3. Add the squares:

   • $36 + 16 = 52$

4. Take the square root:

   • $\sqrt{52} = \sqrt{4 \cdot 13} = 2\sqrt{13}$ (simplified form)
   • Approximation: $\sqrt{52} \approx 7.211$

Final Answer:

The length of the line segment is $2\sqrt{13}$ or approximately $7.21$.

Would you like a step-by-step visualization or have any questions?

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Here are 5 related questions to expand your understanding:

1. How would you calculate the midpoint of the segment joining $C(-2, 1)$ and $D(4, 5)$?
2. What is the slope of the line passing through $C(-2, 1)$ and $D(4, 5)$?
3. If another point $E(1, 2)$ lies on the same line, can you verify it using the slope?
4. What is the equation of the line passing through $C(-2, 1)$ and $D(4, 5)$?
5. How does the distance formula relate to the Pythagorean theorem?

Tip: The distance formula is a direct application of the Pythagorean theorem in a coordinate plane!