Understanding the Relationships and Properties of a 45-45-90 Triangle in Geometry and Trigonometry

45-45-90 triangle

A 45-45-90 triangle is a special type of right triangle where the two smaller angles are each equal to 45 degrees, and the third angle, which is the right angle, measures 90 degrees

A 45-45-90 triangle is a special type of right triangle where the two smaller angles are each equal to 45 degrees, and the third angle, which is the right angle, measures 90 degrees. This triangle is also known as an isosceles right triangle because it has two equal sides.

The sides of a 45-45-90 triangle have specific relationships with each other. Let’s denote the length of the two equal legs as ‘a’ and the length of the hypotenuse as ‘c’.

The most important property of a 45-45-90 triangle is that the two legs are congruent, which means they have equal lengths. Therefore, if ‘a’ represents the length of the legs, both legs will have a length of ‘a’. This property arises because the triangle is isosceles.

The length of the hypotenuse can be found using the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. For a 45-45-90 triangle, we have:

c^2 = a^2 + a^2
c^2 = 2a^2

To find the exact lengths of the sides, we can take the square root of both sides of the equation:

c = √(2a^2)
c = a√2

So, in a 45-45-90 triangle, the length of the hypotenuse is equal to the length of each leg multiplied by the square root of 2. The relationship between the sides can be summarized as:

Leg length: a
Hypotenuse length: a√2

For example, if one leg of a 45-45-90 triangle is 5 units, then the other leg will also be 5 units, and the hypotenuse will be 5√2 units.

This special triangle is frequently used in geometry and trigonometry because it has simple and predictable relationships between its sides.

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