Question

Abby used the law of cosines for TriangleKMN to solve for k.

k2 = 312 + 532 – 2(31)(53)cos(37°)

Triangle K N M is shown. The length of K N is 53, the length of N M is k, and the length of K M is n.

Law of cosines: a2 = b2 + c2 – 2bccos(A)

What additional information did Abby know that is not shown in the diagram?

mAngleK = 37° and n = 31
mAngleK = 37° and k = 31
mAngleN = 37° and n = 31
mAngleN = 37° and k = 31

Answer 1

Answer:

  m∠K = 37° and n = 31

Step-by-step explanation:

A lot of math is about matching patterns. Here, the two patterns we want to match are different versions of the same Law of Cosines relation:

• a² = b² +c² -2bc·cos(A)
• k² = 31² +53² -2·31·53·cos(37°)

Comparison

Comparing the two equations, we note these correspondences:

• a = k
• b = 31
• c = 53
• A = 37°

Comparing these values to the given information, we see that ...

• KN = c = 53 . . . . . . . . . . matching values 53
• NM = a = k . . . . . . . . . . . matching values k
• KM = b = n = 31 . . . . . . . matching values 31
• ∠K = ∠A = 37° . . . . . . . matching side/angle names

Abby apparently knew that ∠K = 37° and n = 31.

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Additional comment

Side and angle naming for the Law of Sines and the Law of Cosines are as follows. The vertices of the triangle are labeled with single upper-case letters. The side opposite is labeled with the same lower-case letter, or with the two vertices at either end.

Vertex and angle K are opposite side k, also called side NM in this triangle.