You get lots of points for answering this and i will mark the first right answer ! (results may change)
i am reasking this question because i really need an answer and no one answered last

O (0,0)

R (k, k)

S (k, 2k)

T (2k, 2k)

U (k, 0)

OT = 2√2 + 2k

Step-by-step explanation:

Since the entire shape is 2k units, the height is 2k.

The length of SU is the height. SU = 2k

SU is made of two lines called SR and RU, both the same length.

If SU = SR + RU and SR = RU, they are half of 2k. SR = RU = k

Other lines that have the same length as RU and SR is OU and ST. They all have the one tick marking which means equal length.

SR = RU = OU = ST = k

We know the coordinates of point O. It is the origin because it's where the x-axis and the y-axis intersect. O (0,0)

Since OU=k, U is k units to the right of O for it's x-coordinate. U is still on the x-axis so it's y-coordinate is 0. U (k, 0)

Since R is k units above U, increase the y-coordinate by k. R (k, k)

Point S is k units above R, increase the y-coordinate by k. S (k, 2k)

Point T is k units to the right of S. Increase the x-coordinate by k, and the y-coordinate does not change. T (2k, 2k)

The length of OT is double the hypotenuse of one of the right triangles.

Use the Pythagorean theorem to find the hypotenuse of triangle STR.

RT² = ST² + SR²

RT² = k² + k²   <=substitute the lengths you know

RT² = 2k²    <=find the square root of both sides

RT = √(2)k

Since triangles STR and ORU are the same, hypotenuse RT is the same as hypotenuse OR.

OT = 2(√(2)k)

OT = 2√2 + 2k