1st Problem: Distance between points $(2,5)$ and $(-1,1)$

The distance formula is given by:

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

Substituting the coordinates $(x_1, y_1) = (2, 5)$ and $(x_2, y_2) = (-1, 1)$:

$d = \sqrt{{(-1 - 2)}^2 + {(1 - 5)}^2}$ $d = \sqrt{{(-3)}^2 + {(-4)}^2}$ $d = \sqrt{9 + 16} = \sqrt{25} = 5$

So, the distance between the points $(2, 5)$ and $(-1, 1)$ is 5 units.

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2nd Problem: Distance between points $(1,5)$ and $(1,0)$

Substituting the coordinates $(x_1, y_1) = (1, 5)$ and $(x_2, y_2) = (1, 0)$ into the distance formula:

$d = \sqrt{{(1 - 1)}^2 + {(0 - 5)}^2}$ $d = \sqrt{0^2 + (-5)^2}$ $d = \sqrt{0 + 25} = \sqrt{25} = 5$

So, the distance between the points $(1, 5)$ and $(1, 0)$ is also 5 units.

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Do you want more details, or have any questions?

Here are five related questions:

1. What is the distance formula derived from?
2. How would you find the midpoint between two points?
3. What is the difference between vertical and horizontal distances on the coordinate plane?
4. Can the distance formula be used in three-dimensional space?
5. How does the Pythagorean theorem relate to the distance formula?

Tip: Always ensure you subtract the coordinates in the correct order to avoid mistakes with negative values!