find the length of the diagonal of the rectangle ABCD where A(1,2) B(1,5) C(3,5) D(3,2)

The length of a diagonal from a rectangle could be
calculated using 2 methods.

The first and easier method is:
the length of a segment when  knowing the coordinates of the endpoints of the
segment.

AC = sqrt[(xC - xA)^2 +
(yC-yA)^2]

Now, we'll substitute the
coordinates:

AC = sqrt [(3-1)^2 +
(5-2)^2]

AC = sqrt
(4+9)

AC = sqrt
(13)

The second method is to calculate the
length and the width of the rectangle and then, using Pythagorean theorem, to calculate
the diagonal.

AB = sqrt[(xB - xA)^2 +
(yB-yA)^2]

AB = sqrt[(1 - 1)^2 +
(5-2)^2]

AB = 3

BC = sqrt[(xC
- xB)^2 + (yC-yB)^2]

BC = sqrt[(3 - 1)^2 +
(5-5)^2]

BC = 2

In the
triangle ABC, where B = 90 degrees, so AC is the hypotenuse, we'll
have:

AC^2 = AB^2 + BC^2

AC^2
= 3^2 + 2^2

AC^2 =
9+4

AC = sqrt
13