Final answer:
The Pythagorean Theorem a² + b² = c² can be used to calculate the length of the hypotenuse c in a right triangle by rearranging it as c = √(a² + b²). An example calculation with legs of 9 and 5 units gives a hypotenuse of approximately 10.3 units. Additionally, trigonometry can find the angle the hypotenuse makes with a reference frame.
Step-by-step explanation:
Understanding the Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in geometry that establishes a relationship between the sides of a right-angled triangle. Specifically, it states that the sum of the squares of the lengths of the legs (a and b) is equal to the square of the length of the hypotenuse (c). That is, a² + b² = c². To find the length of the hypotenuse, you can rearrange this formula and solve for c: c = √(a² + b²).
For example, if the lengths of the legs are 9 and 5 units, you would calculate the hypotenuse as follows: c = √(9² + 5²) = √(81 + 25) = √106. To one decimal place, the hypotenuse is approximately 10.3 units. In addition to calculating the length, you can also use trigonometry to determine the direction of the hypotenuse by measuring the angle it makes with a reference frame. The angle, often denoted θ, can be found using the arctangent function: θ = tan⁻¹(b/a).