To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in a coordinate plane, we can use the distance formula:

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

a) Distance between $(0, 5)$ and $(2, -5)$:

• $(x_1, y_1) = (0, 5)$
• $(x_2, y_2) = (2, -5)$

Now, applying the distance formula:

$\text{Distance} = \sqrt{(2 - 0)^2 + (-5 - 5)^2} = \sqrt{2^2 + (-10)^2} = \sqrt{4 + 100} = \sqrt{104} \approx 10.2$

So, the distance is approximately 10.2 units.

b) Distance between $(-1, -3)$ and $(-3, 2)$:

• $(x_1, y_1) = (-1, -3)$
• $(x_2, y_2) = (-3, 2)$

Applying the distance formula:

$\text{Distance} = \sqrt{(-3 - (-1))^2 + (2 - (-3))^2} = \sqrt{(-3 + 1)^2 + (2 + 3)^2} = \sqrt{(-2)^2 + 5^2} = \sqrt{4 + 25} = \sqrt{29} \approx 5.39$

Thus, the distance is approximately 5.39 units.

Would you like further details or have any questions about the steps?

Here are 5 related questions:

1. How do you calculate the midpoint between two points in the coordinate plane?
2. Can you generalize the distance formula for three-dimensional space?
3. How does the distance formula relate to the Pythagorean theorem?
4. What are the units of the calculated distance between two points?
5. How can you find the slope between two points?

Tip: When using the distance formula, always double-check the subtraction inside the parentheses before squaring.