Distance formula, step by step

Two points anywhere on the plane make a right triangle nobody has drawn yet.

The question

units

(0, −5) to (4, −2)

(0, −5)to(4, −2)

across=4 − 0 = 4

up=−2 − (−5) = 3

4² + 3²=25

distance=5 units

There is no new formula here. Draw the horizontal and vertical steps between the two points and they make a right triangle — the line you were given is its longest side.
Across: 4 − 0 = 4, so that side is 4 long. Up: −2 − (−5) = 3, so that side is 3. A length has no sign, so take both as positive.
Now it is the theorem you already know: 4² + 3² = 16 + 9 = 25.
The distance is the square root of that: √25 = 5. So the two points are 5 units apart.

How it works.

01
There is no new formula here. Draw the horizontal and vertical steps between the two points and they make a right triangle — the line you were given is its longest side.
02
Across: 4 − 0 = 4, so that side is 4 long. Up: −2 − (−5) = 3, so that side is 3. A length has no sign, so take both as positive.
03
Now it is the theorem you already know: 4² + 3² = 16 + 9 = 25.
04
The distance is the square root of that: √25 = 5. So the two points are 5 units apart.