In a right triangle, one of the angle is of 90 degrees. The sum of other two angles will be 90 degrees. Two special right triangles are 45-45-90 and 30-60-90.

The 45-45-90 triangle is formed by cutting the square in half along with the diagonal. This triangle is also called isosceles right triangle as two sides are equal. If you bisect a square with a diagonal line, you get two triangles that both have two 45-degree angles. Its hypotenuse is equal to √2 times the length of a leg. This also means that the length of a leg is equal to the length of the hypotenuse divided by √2. 

The 30-60-90 triangle is formed by cutting equilateral triangle in half. When you bisect any angle in an equilateral triangle, you get two such right triangles. In a 30:60:90-degree triangle, the hypotenuse is 2 times the length of the shorter leg. The ratio of the three sides is a : √3a : 2a, where a is the length of the shortest side.

The speciality of these two triangles are that length of the sides of the triangle are in a certain fixed ratio. The ratio can be found out by using pythagoras theorem.

special right triangles

Pythagorean Theorem

The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle. A right triangle is any triangle that has one right internal angle.

According to this theorem, if the length of the legs (smallest sides) are squared and their sum is found, the sum is equal to the square of the hypotenuse (longest side). The Pythagorean theorem simply states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse.

a2 + b2 = c2

where a and b represent the two legs of the right triangle and c is the hypotenuse.

right-triangle

The most common ratio of the three sides of a right triangle is 3:4:5 (3 is the measure of the short leg, 4 is the measure of the long leg, and 5 is the measure of the hypotenuse). Related multiples are 6:8:10, 9:12:15, and so on. Other proportions of right triangles are 5:12:13, 8:15:17, and 7:24:25.