Question

Find x so the distance between
(x, 2)
and
(1, 4)
is square root 13

Answer 1

The formula of a distance between two points:

`d=√((x_2-x_1)^2+(y_2-y_1)^2)`

We have the points (x, 2) and (1,4) and the distance d = √13.

Substitute:

`√((1-x)^2+(4-2)^2)=√(13)\\\\√((1-x)^2+2^2)=√(13)\\\\√((1-x)^2+4)=√(13)\iff(1-x)^2+4=13\qquad|-4\\\\(1-x)^2=9\iff1-x=\pm\sqrt9\\\\1-x=-3\ \vee\ 1-x=3\qquad|-1\\\\-x=-4\ \vee\ -x=2\qquad|\text{change the signs}\\\\\boxed{x=4\ \vee\ x=-2}`

Answer: x = -2 or x = 4.

Answer 2

distance = sqrt( (x1-x2)^2+(y1-y2)^2)

sqrt(13) = sqrt ((x-1)^2+(4-2)^2)

square each side

13 = ((x-1)^2+(4-2)^2)

13 = ((x-1)^2+(2)^2)

13 = (x-1)^2+4

subtract 4 from each side

9 = (x-1)^2

take the square root of each side (remember plus and minus)

sqrt(9) - sqrt((x-1)^2)

3 = x-1 -3 = x-1

add 1 to each side

4 =x -2 =x

Answer x= -2, 4