Understanding the ratios in a 45-45-90 triangle | Explained with a mathematical proof

Angles= 45-45-90; sides= 1-1-√(2)

The given triangle is an isosceles right triangle, also known as a 45-45-90 triangle

The given triangle is an isosceles right triangle, also known as a 45-45-90 triangle. In this type of triangle, two of the angles are equal at 45 degrees each, and the third angle is a right angle, measuring 90 degrees.

The sides of this triangle are in the ratio 1:1:√2. This means that the lengths of the sides are proportional to each other. In this specific case, each leg of the triangle has a length of 1 unit, and the hypotenuse has a length of √2 units.

To understand why the ratio of the sides is 1:1:√2, let’s consider the Pythagorean theorem. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Let one leg of the triangle be x, and because the triangle is isosceles, the other leg will also be x. The hypotenuse can be denoted as c.

According to the Pythagorean theorem, we have:

x^2 + x^2 = c^2

2x^2 = c^2

Taking the square root on both sides:

√(2x^2) = √(c^2)

√2 * x = c

Therefore, the length of the hypotenuse (c) is equal to √2 times the length of the legs (x), which is why the sides of the triangle are in the ratio 1:1:√2.

In summary, for a 45-45-90 triangle, the sides are in the ratio 1:1:√2. In this case, the lengths of the sides are 1-1-√2, where the legs have a length of 1 unit each, and the hypotenuse has a length of √2 units.

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