The law of cosines for [tex]\triangle R S T[/tex] can be set up as [tex]$5^2=7^2+3^2-2(7)(3) \cos (S)$[/tex]. What could be true about [tex]\triangle R S T[/tex]?
Law of cosines: [tex]$a^2=b^2+c^2-2 b c \cos (A)$[/tex]
A. [tex]$r=5[/tex] and [tex]$t=7[/tex]
B. [tex]$r=3[/tex] and [tex]$t=3[/tex]
C. [tex]$s=7[/tex] and [tex]$t=5[/tex]
D. [tex]$s=5[/tex] and [tex]$t=3[/tex]
Answer (2)
Based on the law of cosines equation provided, the only valid option about triangle RST that fits the conditions is option D: s = 5 and t = 3 . This indicates that side lengths 5 and 3 are opposite the angle S signature, while side length 7 remains as the remaining side. Therefore, option D is the correct choice.
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The problem uses the law of cosines to determine the possible side lengths of a triangle. By comparing the given equation with the general form of the law of cosines, we identify the side opposite the given angle and the other two sides. We then check each option to see if it is consistent with the identified side lengths. The only possible option is s = 5 and t = 3 .
Explanation
Analyze the given information We are given the law of cosines for △ RST as 5 2 = 7 2 + 3 2 − 2 ( 7 ) ( 3 ) cos ( S ) . We want to determine which of the given options could be true about the triangle. The general form of the law of cosines is a 2 = b 2 + c 2 − 2 b c cos ( A ) , where a is the side opposite angle A , and b and c are the other two sides.
Identify the sides and angle Comparing the given equation with the general form, we can see that the side opposite angle S has length 5, and the other two sides have lengths 7 and 3. This means that s = 5 , and the other two sides are r and t (or t and r ), which are 7 and 3 in some order.
Check the given options Now, let's check the given options:
Option 1: r = 5 and t = 7 . This implies s must be either 3, but we know s = 5 , so this is not possible.
Option 2: r = 3 and t = 3 . This implies s = 5 , and the other two sides are both 3. However, we know that the other two sides must be 7 and 3, so this is not possible.
Option 3: s = 7 and t = 5 . This implies the side opposite to angle S is 7, but we know it is 5, so this is not possible.
Option 4: s = 5 and t = 3 . This implies r = 7 . This is possible, as s = 5 and the other two sides are 7 and 3.
Conclusion Therefore, the only option that could be true is s = 5 and t = 3 .
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. In real-world applications, this law is used in surveying, navigation, and engineering to calculate distances and angles when direct measurement is not possible. For example, surveyors use the law of cosines to determine the distance between two points when there is an obstacle in between, or to calculate the angles in a triangular plot of land. Civil engineers use it to design bridges and buildings, ensuring structural stability by accurately determining the forces and angles involved.
