Which of the following is a key element in the proof of the law of cosines?
A. [tex]$d=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}$[/tex]
B. [tex]$A=\frac{1}{2} b_0 h$[/tex]
C. The triangle inequality theorem
D. [tex]$\angle A+\angle B+\angle=180$[/tex]
Answer (1)
The key element in the proof of the law of cosines is:
The law of cosines relates the sides and angles of a triangle.
The distance formula can be used to derive the law of cosines by placing a triangle on the coordinate plane.
The distance formula expresses the lengths of the sides in terms of the coordinates of the vertices.
Therefore, the distance formula is a key element in the proof: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
Explanation
Problem Analysis The question asks us to identify a key element in the proof of the law of cosines from the given options. Let's analyze each option to determine its relevance.
Analyzing the options Option A: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 is the distance formula. The law of cosines can be derived by placing a triangle on the coordinate plane and using the distance formula to express the lengths of the sides in terms of the coordinates of the vertices. This is a common and direct approach.
Option B: A = 2 1 b 0 h is the formula for the area of a triangle. While the area of a triangle is related to the sides and angles of a triangle, it is not directly used in the standard proofs of the law of cosines.
Option C: The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. While this is a fundamental property of triangles, it is not directly used in the proof of the law of cosines.
Option D: ∠ A + ∠ B + ∠ C = 180 states that the sum of the angles in a triangle is 180 degrees. This is a fundamental property of triangles, but it is not directly used in the proof of the law of cosines.
Identifying the key element The distance formula is directly used in a common proof of the law of cosines. By placing a triangle on the coordinate plane, we can use the distance formula to find the lengths of the sides and then relate them to the cosine of an angle using algebraic manipulations.
Conclusion Therefore, the key element in the proof of the law of cosines is the distance formula.
Examples
The Law of Cosines is a fundamental concept in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It's incredibly useful in fields like surveying, navigation, and engineering. For instance, surveyors use the Law of Cosines to calculate distances and angles in irregular land plots, helping them create accurate maps and property boundaries. Imagine you're mapping a triangular plot of land where you know the lengths of two sides and the angle between them; the Law of Cosines allows you to determine the length of the third side, completing your survey.
