Special Right Triangle

Right Triangle 

The right triangle, or the right-angled triangle, is one of the basic geometric shapes widely used in mathematics and various fields of science. It is a triangle with one angle equal to 90 degrees or a right angle, separating the triangle into two shorter legs and a longer hypotenuse. 

The concept of the right triangle holds immense significance in trigonometry, geometry, and calculus. In this essay, we will discuss the properties of the right triangle and its applications.  

The Pythagorean Theorem is a famous mathematical principle that applies to right triangles. For any right triangle, if you add together both legs squared, it will be equivalent to the length of the hypotenuse squared. This can be written mathematically as: a2+ b2= c2

Where a and b are the lengths of the legs, and c is the length of the hypotenuse.  

Right triangles are commonly used in geometry and trigonometry, and they have many practical applications in fields such as engineering, physics, and architecture.  

Special Right Triangles  

A special Right triangle is a triangle that has one of its angles measuring 90 degrees (a Right angle) and two of its sides having a special relationship with each other. 

For example, the two special right triangles are the 45–45–90 right triangle and the 30–60–90 right triangle. These triangles have unique properties that make them easier to work with than regular ones.  

The 45–45–90 right triangle has two sides of equal length and one side longer than the other two sides. The ratio of the sides in a 45–45–90 right triangle is x:x:x√2, where x is the length of the shorter side, and x√2 is the length of the longer side.

 This can be derived from the Pythagorean theorem, which states that the square of the hypotenuse equals the sum of squares of the other two sides.   

The 30–60–90 right triangle has two legs of different lengths and one hypotenuse, where the length of the hypotenuse is double the length of the shorter leg. The ratio of the sides in a 30–60–90 right triangle is x:x√3:2x, where x is the length of the shorter leg.  

One of the unique properties of the special right triangles is that the lengths of the sides can be easily found if one of the sides is known. 

For instance, if one side of a 45–45–90 right triangle is known, the remaining sides can be found by multiplying by the square root of two. Similarly, if one side of a 30–60–90 right triangle is known, the other side can be found by multiplying it by the square root of three or dividing by two to find the hypotenuse.  

Special Right Triangle Formula  

45-45-90 triangle:  

In a 45-45-90 triangle, the two legs are congruent, and the hypotenuse is √2 times the length of a leg.  

So, if the length of one leg is "a," then:  

Length of the other leg = a  

Length of the hypotenuse = a√2  

30-60-90 triangle:  

In a 30-60-90 triangle, the length of the longer leg is √3 times the length of the shorter leg, and the length of the hypotenuse is twice that of the shorter leg.  

Consider the length of the shorter leg as "a," then:  

Length of the longer leg = a√3  

Length of the hypotenuse = 2a