Distance between two points, step by step

There is no new theorem on these sheets — only a new place to notice the old one.

The question

units

(5, 7) to (9, 4)

(5, 7)to(9, 4)

across=9 − 5 = 4

up=4 − 7 = −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: 9 − 5 = 4, so that side is 4 long. Up: 4 − 7 = −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: 9 − 5 = 4, so that side is 4 long. Up: 4 − 7 = −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.