Question

question 7 of 10
suppose a triangle has sides a, b, and c, and the angle opposite the side of length b is obtuse. what must be true?
a. ( a^{2}+b^{2}<c^{2} )
b. ( b^{2}+c^{2}<a^{2} )
c. ( a^{2}+c^{2}<b^{2} )
d. ( a^{2}+c^{2}>b^{2} )

Explanation:

Step1: Recall the Law of Cosines

The Law of Cosines for a triangle with sides \(a\), \(b\), \(c\) and the angle \(B\) opposite side \(b\) is given by \(b^{2}=a^{2}+c^{2}-2ac\cos B\).

Step2: Analyze the case when angle \(B\) is obtuse

If angle \(B\) is obtuse, then \(\cos B< 0\) (since cosine of an obtuse angle, which is between \(90^{\circ}\) and \(180^{\circ}\), is negative).
Substitute \(\cos B\) into the Law of Cosines formula: \(b^{2}=a^{2}+c^{2}-2ac\cos B\). Since \(\cos B < 0\), the term \(- 2ac\cos B\) will be positive (because a negative times a negative is positive). So, \(b^{2}=a^{2}+c^{2}+\text{(positive term)}\). This implies that \(a^{2}+c^{2}

Answer:

C. \(a^{2}+c^{2}