
Right Triangles And Pythagorean Theorem 
The Pythagorean Theorem states that in a
right triangle, the square of the hypotenuse is equal to the sum
of the squares of the legs.
The hypotenuse is the longest side of a right triangle. It is always across from the right angle.
Based on this right triangle, you can write an equation using the Pythagorean Theorem as follows:


a^{2} + b^{2} = c^{2}




Pythagorean triple 
A Pythagorean triplet is made up of three natural numbers, which may appear as lengths of the sides of a right triangle. 
 A Pythagorean triple is a right triangle whose sides are in the ratio 3:4:5
 The side lengths do not
need to measure 3, 4, and 5; however, they do need to reduce to that ratio
 For example, the side
lengths 9, 12, and 15 are a Pythagorean triple because it simplifies to the ratio 3:4:5


The Converse of the Pythagorean Theorem 
You can also use the Pythagorean Theorem to determine if a triangle is acute, right, or obtuse. This is known as the converse of the Pythagorean Theorem, which reads as follows: 
 If c^{2} < a^{2}+ b^{2}, then the triangle is acute
 If c^{2} = a^{2} + b^{2}, then the triangle is right
 If c^{2} > a^{2} + b^{2}, then the triangle is obtuse


Special Right Triangles 
Two types of right triangles are considered special right triangles. One of the special right triangles has angles that measure 30°, 60°, and 90°. The other special right triangle has angles that measure 45°, 45°, and 90°.
The size of the triangle does not matter; it just needs to have specific measures for its angles. 

30°60°90° Triangle 
The lengths of the sides of a 30°60°90° triangle are in a ratio of 

The hypotenuse is twice the length of the side opposite the30° angle. 
 Therefore, a must
be the following: a = c/2
 And b must be:

Using this information, you can determine the value of a and b for the triangle, if you know the value of c. 

45°45°90° Triangle 
The sides a of a 45°  45°  90° triangle are the same length. 

 The ratio of side lengths to hypotenuse is
 This results in


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