In a rectangle ABCD ,diagonals AC and BD bisected and AC=10cm. What is the length of AD,AE,AB?

Since the diagonals AC and BD are bisected, they divide the rectangle into four congruent right triangles.

Therefore, the length of AD, AE, and AB can be calculated using the Pythagorean theorem in one of the right triangles.

Let x be the length of AD, AE, and AB.

In triangle ADE:
AD^2 + AE^2 = DE^2
x^2 + x^2 = (10/2)^2
2x^2 = 25
x^2 = 25/2
x = √(25/2)
x = √(25)/√(2)
x = 5/√(2)
x = 5√(2)/2

Therefore, the length of AD, AE, and AB is 5√(2)/2 cm.

Since the diagonals of a rectangle bisect each other, we have:

AE = EC = AC/2 = 10/2 = 5 cm

Therefore, AD = AE = 5 cm.

Also, since the diagonals of a rectangle are perpendicular bisectors of each other, we have:

AB^2 = AE^2 + EB^2

Substituting EB = EC = 5 cm, we get:

AB^2 = 5^2 + 5^2
AB^2 = 25 + 25
AB^2 = 50
AB = √50 = 5√2 cm

So, the lengths of AD, AE, and AB are 5 cm, 5 cm, and 5√2 cm, respectively.