Suppose a triangle has sides [tex]a, b[/tex], and [tex]c[/tex], and that [tex]a^2+b^2\ \textgreater \ c^2[/tex]. Let [tex]\theta[/tex] be the measure of the angle opposite the side of length [tex]c[/tex]. Which of the following must be true? Check all that apply.
A. The triangle is a right triangle.
B. [tex]\theta[/tex] is an acute angle.
C. The triangle is not a right triangle.
D. [tex]cos \theta\ \textgreater \ 0[/tex]
Answer (1)
Apply the Law of Cosines to relate the sides of the triangle to the cosine of the angle opposite side c.
Use the given condition c^2"> a 2 + b 2 > c 2 to show that 0"> cos θ > 0 .
Deduce that θ is an acute angle because 0"> cos θ > 0 .
Conclude that the triangle is not a right triangle. B, C, D
Explanation
Problem Setup and Given Information We are given a triangle with sides a , b , c and the condition c^2"> a 2 + b 2 > c 2 . We want to determine which statements about the angle θ opposite side c must be true.
Applying the Law of Cosines We can use the Law of Cosines, which states that c 2 = a 2 + b 2 − 2 ab cos θ . We can rearrange this to solve for cos θ :
Isolating cos θ cos θ = 2 ab a 2 + b 2 − c 2
Analyzing the Sign of the Numerator and Denominator Since we are given that c^2"> a 2 + b 2 > c 2 , it follows that 0"> a 2 + b 2 − c 2 > 0 . Also, a and b are side lengths of a triangle, so 0"> a > 0 and 0"> b > 0 . Therefore, 0"> 2 ab > 0 .
Determining the Sign of cos θ Since both the numerator ( a 2 + b 2 − c 2 ) and the denominator ( 2 ab ) of the expression for cos θ are positive, we have 0"> cos θ > 0 .
Determining the Nature of Angle θ If 0"> cos θ > 0 , then θ must be an acute angle (i.e., 0 < θ < 9 0 ∘ ).
Triangle Type If θ is an acute angle, then the triangle cannot be a right triangle.
Final Answer Therefore, the following statements must be true: B. θ is an acute angle. D. 0"> cos θ > 0 C. The triangle is not a right triangle.
Examples
In architecture, when designing roof structures, it's crucial to ensure that the angles formed by the supporting beams are stable. If you know the lengths of the beams and want to ensure that the angle opposite the longest beam is acute, you can use the condition c^2"> a 2 + b 2 > c 2 to verify this, where a and b are the lengths of the shorter beams and c is the length of the longest beam. This helps in creating a structurally sound and aesthetically pleasing roof design.
