What is the distance between `(1,-4)` and `(7,5)`?

make a right angle triangle with the two points being the end points of the hypotenuse.

The distance between the `x` values is 7-1=6

The distance between the `y` values is 5- -4=5+4=9

So our triangle has two shorter sides 6 and 9 and we need to find the length of the hypotenuse, use Pythagoras.

`6^2+9^2=h^2`

`36+81+117`

`h=sqrt117=3sqrt13`

`"calculate the distance d using the "color(blue)"distance formula"`

`•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`"let "(x_1,y_1)=(1,-4)" and "(x_2,y_2)=(7,5)`

`d=sqrt((7-1)^2+(5-(-4))^2)`

`color(white)(d)=sqrt(6^2+9^2)=sqrt(36+81)=sqrt117~~10.82`

If you were to draw a right triangle so that the hypotenuse is the line between `(1,-4)` and `(7,5)`, you would observe that the two legs of the triangle would be of length `6` (i.e. the distance between `x=7` and `x=1`) and `9` (i.e. the distance between `y=5` and `y=-4`). By applying the pythagorean theorem,

`a^2+b^2=c^2`,

where `a` and `b` are the lengths of the legs of a right triangle and `c` is the length of the hypotenuse, we obtain:

`6^2 + 9^2 = c^2`.

Solving for the length of the hypotenuse (i.e. the distance between the points `(1,-4)` and `(7,5)`), we get:

`c=root()117`.

The process of finding the distance between two points by use of a right triangle can be formulated thusly:

Distance`= root()((x_2−x_1)^2+(y_2−y_1)^2)`.

This is called the distance formula, and can be used to expedite the solving of this sort of problem.