Hi Aaditya,

The relationship between the lengths of the sides of a 45-45-90 triangle can be determined by examining the triangle formed by drawing the diagonal in a unit square (a square whose sides are all of length 1 unit).

The diagonal bisects the angles whose vertices it joins, and the square is now two 45-45-90 triangles with the sides of the original square the legs and the diagonal the hypotenuse.

The legs of this triangle are each of length 1 unit as they are both original sides of the unit square. That is, the legs of this 45-45-90 triangle are equal.

Applying the Pythagorean theorem we find that the length of the hypotenuse is equal to the square root of 2. In other words, the hypotenuse is root 2 times the length of either leg.

If this triangle is "scaled up" or "scaled down" to make the lengths of sides and hypotenuse larger or smaller than 1 unit, the leg to leg ratio of 1 to 1 and hypotenuse to leg ratio of square root 2 to 1 do not change.

In other words, in every 45-45-90 triangle, the lengths of the two legs are always equal, and the ratio of the length of the hypotenuse to the length of a leg is always square root 2 to 1.

So if one leg of a 45-45-90 triangle is 3, then the other leg is also 3, and the hypotenuse must be 3 times the square root of 2 in order to maintain the ratio.

Hope this helps,
Leeanne