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A right triangle (a triangle with one 90-degree angle) with two 45-degree angles is known as a 45-45-90 triangle. Due to its distinctive qualities, this particular kind of right triangle is important in geometry and mathematics.

A 45-45-90 triangle has the characteristic that the ratio of the lengths of the sides is always 1:1:√2. In other words, if one of the triangle's sides, or its "leg," is "a," then its other leg is also "a," and its hypotenuse, or the side that faces the right angle, is "√2a."

Each of the triangle's three angles measures 45 degrees, which is another aspect of a 45-45-90 triangle. Therefore, this triangle is always an isosceles triangle (a triangle that has two sides of equal length).

An isosceles right triangle or a 45-degree right triangle are just other names for a 45-45-90 triangle. It can be used to resolve problems involving right triangles and to derive other significant mathematical relationships, making it a useful shape in mathematics

In the case of a 45-45-90 triangle, the length of the hypotenuse can be resolved by using the Pythagorean theorem (a2 + b2 = c2) if the leg lengths are known. To find the hypotenuse's length, the theorem can be rearranged as follows: c = √(a2 + b2). The theorem becomes c = √(2a2) = √2a because the legs of a 45-45-90 triangle are equal in length (a = b). This demonstrates that the hypotenuse of a 45-45-90 triangle is approximately √2 as long as each leg.

To find the length of the hypotenuse of a 45-45-90 triangle with sides of length 6, you can use the Pythagorean theorem.

The Pythagorean theorem can be used to calculate the length of the hypotenuse of a triangle measuring 45, 45, and 90 degrees and with sides of length 6.

The Pythagorean theorem mentions that in a right triangle, the square of the length of the hypotenuse (the side that is opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (the legs). You can write this as a^2 + b^2 = c^2, where a and b are the lengths of the legs and c is the length of the hypotenuse.

In a 45-45-90 triangle, the two legs are always equal in length, so you can let a = b = 6. Plugging these values into the Pythagorean theorem gives you:

a^2 + b^2 = c^2

6^2 + 6^2 = c^2

36 + 36 = c^2

72 = c^2

To determine the value of c, take the square root of both sides of the equation:

√(72) = √(c^2)

8.485281374238571 = c

Therefore, the hypotenuse of a triangle measuring 45, 45, and 90 with sides of length 6 is about 8.49 units long.