i need to figure out side lengths of congruent right isosceles triangle with no lengths given. I know that the hypotenese is 1 per pythogoras constant and the area is a^2/2. I don't have to prove area just side lengths.

To find the side lengths of a congruent right isosceles triangle, we can use the Pythagorean theorem and the knowledge that the hypotenuse is equal to 1 (per Pythagoras constant).

Let's denote the length of each side of the triangle as 's'. Since it is a right isosceles triangle, two sides will have length 's' and the remaining side, the hypotenuse, will have length 1.

Using the Pythagorean theorem, we have:

s^2 + s^2 = 1^2

Simplifying the equation:

2s^2 = 1

Dividing both sides by 2, we get:

s^2 = 1/2

To find the side length 's', we take the square root of both sides:

s = √(1/2)

Now, simplifying the square root:

s = √(1) / √(2)

Since the square root of 1 is 1:

s = 1 / √(2)

To rationalize the denominator, we multiply the numerator and denominator by the conjugate of √(2) which is also √(2):

s = (1 / √(2)) * (√(2) / √(2))

Multiplying the numerators and denominators, we get:

s = √(2) / (2)

So, the side length of the congruent right isosceles triangle is √(2) / (2).

Please note that this calculation assumes that the triangle is scaled such that its hypotenuse is equal to 1.