Given triangle $KNM$, which of the following statements about its sides is true?

• A. $KN+NM=KM$
• B. $KM=2\left(NM\right)$
• C. The Law of Cosines relates the sides as $K^M=K^N+N^M-2KNNM\cos \angle K$
• D. $KN=NM$

Answer: (C)

Verified step by step guidance

1. Identify the triangle and label its sides and angles clearly. Let the sides be \( KN, NM, \) and \( KM \), and the angle opposite side \( KM \) be \( \angle K \).
2. Recall the Law of Cosines formula, which relates the lengths of the sides of a triangle to the cosine of one of its angles: \[ \text{side}^2 = \text{adjacent side}_1^2 + \text{adjacent side}_2^2 - 2 \times (\text{adjacent side}_1) \times (\text{adjacent side}_2) \times \cos(\text{included angle}) \] In this case, it is: \[ KM^2 = KN^2 + NM^2 - 2 \times KN \times NM \times \cos(\angle K) \]
3. Analyze the given statements: - The first statement \( KN + NM = KM \) suggests the triangle inequality is an equality, which only happens if the points are collinear, not a triangle. - The second statement \( KM = 2(NM) \) is a specific length relation that may or may not hold. - The third statement is the Law of Cosines formula, which is always true for any triangle. - The fourth statement \( KN = NM \) states the two sides are equal, which is a special case (isosceles triangle) but not generally true.
4. Conclude that the Law of Cosines formula correctly relates the sides and angle of the triangle in all cases, making it the true and general statement about the sides of triangle \( KNM \).
5. To verify or use the Law of Cosines, you would substitute the known side lengths and angle measure into the formula: \[ KM^2 = KN^2 + NM^2 - 2 \times KN \times NM \times \cos(\angle K) \] and solve for the unknown side or angle as needed.