In the study of right triangles, the 45-45-90 triangle is a special case that frequently appears in various mathematical contexts. This triangle is characterized by having two angles of 45 degrees and one right angle, which results in the two legs being of equal length. This property allows for efficient calculations when determining the lengths of the sides.
To find the hypotenuse of a 45-45-90 triangle, one can use the shortcut that involves multiplying the length of one leg by the square root of 2. For instance, if each leg measures 5 units, the hypotenuse can be calculated as:
\[ c = a \sqrt{2} \]
where \( a \) is the length of a leg. Thus, the hypotenuse would be \( 5 \sqrt{2} \).
Alternatively, the Pythagorean theorem can be applied, which states that the sum of the squares of the legs equals the square of the hypotenuse:
\[ a^2 + b^2 = c^2 \]
In this case, if both legs are 5, the equation becomes:
\[ 5^2 + 5^2 = c^2 \]
Calculating this gives:
\[ 25 + 25 = c^2 \]
\[ 50 = c^2 \]
Taking the square root of both sides results in:
\[ c = \sqrt{50} = 5\sqrt{2} \]
This confirms that both methods yield the same result, but the shortcut is significantly quicker.
When solving for unknown sides in a 45-45-90 triangle, if one leg is known, simply multiply it by the square root of 2 to find the hypotenuse. Conversely, if the hypotenuse is known, divide it by the square root of 2 to find the length of each leg. For example, if the hypotenuse is 13, the calculation would be:
\[ \text{Leg} = \frac{13}{\sqrt{2}} \]
To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{2} \):
\[ \text{Leg} = \frac{13\sqrt{2}}{2} \]
Thus, both legs of the triangle would measure \( \frac{13\sqrt{2}}{2} \).
Understanding these properties and shortcuts for 45-45-90 triangles not only simplifies calculations but also enhances problem-solving efficiency in geometry.
