Methods for Finding the Hypotenuse Length

The fundamental method for determining the length of the hypotenuse in a right-angled triangle, when the lengths of the other two sides (the legs) are known, is to Use the Pythagorean Theorem.
This theorem is a cornerstone of Euclidean geometry and provides a definitive relationship between the three sides of a right triangle.

The Pythagorean Theorem states that in any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs.
Verbatim Definition: The relationship is mathematically expressed as: $a^2 + b^2 = c^2$
Variable Definitions: - $a$ : The length of one leg of the right triangle. - $b$ : The length of the second leg of the right triangle. - $c$ : The length of the hypotenuse (the side opposite the ninety-degree angle).

To find the length of the hypotenuse ( $c$ ) using the theorem, one must follow these precise steps:

Step 1: Squaring the Legs: Calculate the square of each leg. If the legs have lengths $a$ and $b$ , find $a^2$ and $b^2$ .
Step 2: Summation: Add the result of the squares together: $a^2 + b^2$ .
Step 3: Finding the Principal Square Root: Since the sum equals $c^2$ , the length of the hypotenuse itself is the square root of that sum: $c = \sqrt{a^2 + b^2}$

The transcript identifies several incorrect methods that do not yield the length of the hypotenuse:

Multiply the lengths of the legs: Multiplying the legs ( $a \times b$ ) is a component of finding the area of the triangle ( $\text{Area} = \frac{1}{2} \times a \times b$ ), but it does not equate to the hypotenuse length.
Add the lengths of the legs: Adding the lengths ( $a + b$ ) is part of the Triangle Inequality Theorem, which indicates that $a + b > c$ . However, the sum of the legs is always strictly greater than the hypotenuse, not equal to it.
Divide the length of the legi: Dividing the length of one leg by another relates to trigonometric functions (such as $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ ), but the quotient does not describe the length of the hypotenuse. (Note: The term "legi" is captured verbatim from the transcript).

The transcript makes a brief reference to "DELL," which appears to be context-specific branding or a hardware reference present during the presentation of the geometric problem on Page 1.