key term - Hypotenuse

Definition

The hypotenuse is the longest side of a right-angled triangle, which is the side opposite the right angle. It is a fundamental concept in trigonometry and is used extensively in the study of vector addition and subtraction through graphical methods.

5 Must Know Facts For Your Next Test

1. The hypotenuse is the longest side of a right-angled triangle and is always opposite the right angle.
2. The Pythagorean Theorem can be used to calculate the length of the hypotenuse if the lengths of the other two sides are known.
3. Trigonometric ratios, such as sine, cosine, and tangent, are defined using the lengths of the sides of a right-angled triangle, with the hypotenuse being a key component.
4. In vector addition and subtraction using graphical methods, the hypotenuse of the resulting vector triangle represents the magnitude of the vector sum or difference.
5. The length of the hypotenuse is a crucial factor in determining the direction and magnitude of the resultant vector when adding or subtracting vectors graphically.

Review Questions

• Explain how the hypotenuse is used in the graphical addition and subtraction of vectors.
  • In the graphical method of vector addition and subtraction, the hypotenuse of the resulting vector triangle represents the magnitude of the vector sum or difference. The length of the hypotenuse is a crucial factor in determining the direction and magnitude of the resultant vector, as it is the longest side of the right-angled triangle formed by the vector components. By using the properties of the hypotenuse, such as its relationship to the other sides of the triangle, the magnitude and direction of the resultant vector can be determined.
• Describe the relationship between the hypotenuse and the Pythagorean Theorem in the context of vector addition and subtraction.
  • The Pythagorean Theorem, which states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides, is a fundamental principle used in the graphical addition and subtraction of vectors. When adding or subtracting vectors using the graphical method, the resulting vector triangle formed has a hypotenuse that represents the magnitude of the vector sum or difference. By applying the Pythagorean Theorem to this triangle, the length of the hypotenuse can be calculated from the lengths of the other two sides, which are the vector components. This relationship allows for the determination of the magnitude and direction of the resultant vector in vector addition and subtraction.
• Analyze how the properties of the hypotenuse, such as its relationship to the trigonometric ratios, are utilized in the graphical methods of vector addition and subtraction.
  • The hypotenuse of a right-angled triangle is a key component in the definition of trigonometric ratios, such as sine, cosine, and tangent. In the context of vector addition and subtraction using graphical methods, the properties of the hypotenuse and its relationship to these trigonometric ratios are extensively utilized. The length of the hypotenuse, along with the lengths of the other two sides, allows for the determination of the magnitude and direction of the resultant vector through the application of trigonometric functions. By understanding how the hypotenuse relates to the trigonometric ratios, one can effectively analyze and interpret the graphical representation of vector operations, ultimately leading to a deeper comprehension of the underlying principles of vector addition and subtraction.

Related terms

Right Angle: A right angle is an angle that measures exactly 90 degrees, or one-quarter of a complete circle.

Pythagorean Theorem:

The Pythagorean Theorem states that in a right-angled 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.

Trigonometric Ratios: The trigonometric ratios, such as sine, cosine, and tangent, are defined using the lengths of the sides of a right-angled triangle, with the hypotenuse being a key component.