Question

Given JM = 27, ML = 16, JL = 46, NK = 15, m∠KLM = 48°, m∠JKM = 78°, and m∠MJL = 22°, find each missing value. *

Answer 1

Using the Law of Cosines, we can find the length of JK:

JK^2 = JM^2 + KM^2 - 2 * JM * KM * cos(m∠JKM)
JK^2 = 27^2 + KM^2 - 2 * 27 * KM * cos(78°)

Using the Law of Cosines again, we can find the length of KL:

KL^2 = KM^2 + ML^2 - 2 * KM * ML * cos(m∠KLM)
KL^2 = KM^2 + 16^2 - 2 * KM * 16 * cos(48°)

Now we can set the two expressions for JK^2 equal to each other and solve for KM:

27^2 + KM^2 - 2 * 27 * KM * cos(78°) = JK^2 = KL^2 = KM^2 + 16^2 - 2 * KM * 16 * cos(48°)
27^2 - 16^2 = 2 * KM * (16 * cos(48°) - 27 * cos(78°))
KM = (27^2 - 16^2) / (2 * (16 * cos(48°) - 27 * cos(78°)))
KM ≈ 18.33

Now we can use the Law of Sines to find the length of JL:

JL / sin(m∠KLM) = KL / sin(m∠MJL)
JL / sin(48°) = KL / sin(22°)
JL = sin(48°) * KL / sin(22°)
JL ≈ 29.52

Therefore, KM ≈ 18.33, JL ≈ 29.52.