Math CentralQuandaries & Queries


Question from Aaditya, a student:

explain to me please how to do the 45-45-90 theorem when one of the legs(not the
hypotenuse) is 3. How do you find the remaining two sides? please help me out.

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,

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