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]$A=\frac{1}{2} b \cdot h$[/tex]
C. [tex]$\angle A+\angle B+\angle=180$[/tex]
D. The triangle inequality theorem
Answer (2)
The key element in the proof of the Law of Cosines is the distance formula, expressed as d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 . This formula is used to calculate distances in coordinate geometry, which helps derive the Law of Cosines. Therefore, the correct answer is Option A.
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The Law of Cosines relates the sides and angles of a triangle: c 2 = a 2 + b 2 − 2 ab cos ( C ) .
The distance formula, d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 , is a key element in proving the Law of Cosines using coordinate geometry.
Other options, such as the area of a triangle and the sum of angles in a triangle, are not directly used in the proof.
Therefore, the correct answer is A: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
Explanation
Analyze the options and the Law of Cosines. The question asks us to identify the key element in the proof of the law of cosines from the given options.
The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is given by the formula: c 2 = a 2 + b 2 − 2 ab cos ( C )
Let's analyze each option:
A. d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 is the distance formula. This formula is used to calculate the distance between two points in a coordinate plane and is indeed a key element in one of the proofs of the Law of Cosines.
B. A = 2 1 b ⋅ h is the formula for the area of a triangle. While important in trigonometry, it's not directly used in proving the Law of Cosines.
C. ∠ 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 a key element in proving the Law of Cosines itself.
D. 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 remaining side. While important in geometry, it's not a direct component in the proof of the Law of Cosines.
Therefore, the distance formula is a key element in the proof of the law of cosines.
Explain the role of the distance formula. The distance formula, d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 , is used in a coordinate geometry proof of the Law of Cosines. By placing a triangle on the coordinate plane and using the distance formula to find the lengths of the sides, we can derive the Law of Cosines.
State the final answer. The key element in the proof of the law of cosines is the distance formula.
Therefore, the answer is A.
Examples
The Law of Cosines is incredibly useful in navigation and surveying. Imagine you're a surveyor trying to determine the distance across a lake. You can measure the distances from your position to two points on opposite sides of the lake, as well as the angle between those lines of sight. Using the Law of Cosines, you can then calculate the distance across the lake without needing to physically measure it. This principle is also used in GPS technology to calculate distances between satellites and receivers, helping you navigate your way around town!
