Suppose an isosceles triangle [tex]A B C[/tex] has [tex]A=\frac{\pi}{4}[/tex] and [tex]b=c=3[/tex]. What is the length of [tex]a^2[/tex] ?
A. [tex]3^2 \sqrt{2}[/tex]
B. [tex]3^2(\sqrt{2}-2)[/tex]
C. [tex]3^2(2-\sqrt{2})[/tex]
D. [tex]3^2(2+\sqrt{2})[/tex]
Answer (1)
Apply the Law of Cosines: a 2 = b 2 + c 2 − 2 b c cos ( A ) .
Substitute the given values: a 2 = 3 2 + 3 2 − 2 ( 3 ) ( 3 ) cos ( 4 π ) .
Simplify the expression using cos ( 4 π ) = 2 2 : a 2 = 18 − 18 ( 2 2 ) = 18 − 9 2 .
Factor the result: a 2 = 3 2 ( 2 − 2 ) .
3 2 ( 2 − 2 )
Explanation
Problem Analysis We are given an isosceles triangle A BC with angle A = 4 π and sides b = c = 3 . We want to find the length of a 2 , where a is the side opposite to angle A .
Applying the Law of Cosines We can use the Law of Cosines to relate the sides and angles of the triangle. The Law of Cosines states that: a 2 = b 2 + c 2 − 2 b c cos ( A ) In our case, we have b = c = 3 and A = 4 π .
Substitution Substitute the given values into the Law of Cosines: a 2 = 3 2 + 3 2 − 2 ( 3 ) ( 3 ) cos ( 4 π ) a 2 = 9 + 9 − 18 cos ( 4 π )
Using the value of cosine We know that cos ( 4 π ) = 2 2 . Substitute this value into the equation: a 2 = 18 − 18 ( 2 2 ) a 2 = 18 − 9 2
Simplification Factor out 9 from the expression: a 2 = 9 ( 2 − 2 ) We can also write this as: a 2 = 3 2 ( 2 − 2 )
Final Answer Therefore, the length of a 2 is 3 2 ( 2 − 2 ) .
Examples
The Law of Cosines, used here to find the side length of a triangle, is also essential in navigation and surveying. For example, surveyors use it to calculate distances and angles in irregular land plots, while sailors and pilots use it to determine their position and course by referencing known landmarks or celestial bodies. Understanding these relationships allows for precise measurements and planning in various real-world scenarios.
