The length of LN is approximately LN.
To calculate the length of LN, we can use the Law of Cosines to find the length of KM. Then, we can use that length to determine the length of LN.
KL = 8.9 cm
LM = 8.8 cm
KM = 7.1 cm
Size of angle NKL = x
Size of angle KLM = 5
Let's denote the length of LN as y.
Applying the Law of Cosines to triangle KLM, we have:
KM² = KL² + LM² - 2(KL)(LM)cos(KLM)
Substituting the given values, we get:
(7.1)² = (8.9)² + (8.8)² - 2(8.9)(8.8)cos(5)
49.41 = 79.21 + 77.44 - 2(8.9)(8.8)cos(5)
49.41 = 156.65 - 2(8.9)(8.8)cos(5)
Now, let's calculate the value of cos(5) using a scientific calculator:
cos(5) ≈ 0.99619
49.41 = 156.65 - 2(8.9)(8.8)(0.99619)
49.41 = 156.65 - 155.848096
49.41 + 155.848096 = 156.65
205.258096 = 156.65
Next, let's use the Law of Sines to relate the lengths of LM, LN, and the angles NKL and KLM:
sin(KLM) / LN = sin(NKL) / LM
sin(5) / LN = sin(x) / 8.8
Now, substitute the values:
sin(5) / LN = sin(x) / 8.8
sin(x) = (sin(5) * 8.8) / LN
Using a scientific calculator, we find:
sin(x) ≈ (0.08716 * 8.8) / LN
sin(x) ≈ 0.766208 / LN
Now, let's solve for LN:
LN ≈ (0.766208) / (sin(x))
Finally, substitute the value of sin(x) we obtained earlier:
LN ≈ (0.766208) / (sin(x))
Substituting the value of sin(x) and rounding the answer to 3 significant figures, we get:
LN ≈ (0.766208) / (0.766208 / LN) ≈ LN
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