Question

Find the length of side x in simplest radical form with a rational denominator. The angles are 45-45-90 while the numbers /3 and x.

Answer 1

I need one detail from the diagram: is the given number on a leg or on the hypotenuse, and is that number 3 or √3? Can you confirm or upload the picture?

Common cases for a 45-45-90 (isosceles right) triangle — let L = length of a leg, H = hypotenuse, x the unknown:

- Relationship: H = L√2 and L = H/√2 = H√2/2 (rationalized).

So depending on which side is given:

- If the hypotenuse H = 3, then each leg x = 3/√2 = (3√2)/2.
- If the hypotenuse H = √3, then each leg x = √3/√2 = √6/2.
- If a leg L = 3 and x is the hypotenuse, x = 3√2.
- If a leg L = √3 and x is the hypotenuse, x = √3·√2 = √6.
- If x is the other leg and one leg is given, then x equals that given leg (legs are equal).

Tell me which of these matches your diagram and I’ll confirm the simplest-radical form.