Question

determining a side using the law of cosines

in △def, de = 11, ef = 9, and angle e = 140°. which equation correctly uses the law of cosines to solve for the third side?

( e^2 = 11^2 + 9^2 - 2(11)(9) cos(140^circ) )

( e = 11 + 9 - 2(11)(9) cos(140^circ) )

Explanation:

Step1: Recall Law of Cosines

The Law of Cosines states that for a triangle with sides \(a\), \(b\), and \(c\), and the angle \(C\) opposite side \(c\), the formula is \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\).

Step2: Identify sides and angle in \(\triangle DEF\)

In \(\triangle DEF\), we have \(DE = 11\), \(EF=9\), and angle \(E = 140^{\circ}\). The side opposite angle \(E\) is \(DF\) (which is labeled as \(e\)). So, \(a = 11\), \(b = 9\), and \(C=140^{\circ}\), and the side we want to find is \(e\) (opposite angle \(E\)).

Step3: Apply Law of Cosines

Substituting into the Law of Cosines formula \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\), we get \(e^{2}=11^{2}+9^{2}-2(11)(9)\cos(140^{\circ})\). The second option is incorrect because it does not square the sides \(11\) and \(9\) as required by the Law of Cosines.

Answer:

\(e^{2}=11^{2}+9^{2}-2(11)(9)\cos(140^{\circ})\) (the first equation in the provided options)