This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To find the length of MN, we will use trigonometry in the two right-angled triangles shown in the figure: KNL and LNM. Step 1: Find the length of LN using KNL. In KNL, we are given: • KNL = 90^ (indicated by the square symbol at N) • KL = 8.5 cm (hypotenuse) • K = 30^ We want to find the length of LN, which is the side opposite to K. Using the sine function: () = oppositehypotenuse (30^) = (LN)/(KL) Substitute the given values: (30^) = (LN)/(8.5 cm) We know that (30^) = 0.5: 0.5 = (LN)/(8.5 cm) Solve for LN: LN = 0.5 × 8.5 cm LN = 4.25 cm Step 2: Find the length of MN using LNM. In LNM, we are given: • NLM = 90^ (indicated by the square symbol at L) • LN = 4.25 cm (calculated in Step 1) • LNM = 55^ We want to find the length of MN, which is the hypotenuse of LNM. LN is the side adjacent to LNM. Using the cosine function: () = adjacenthypotenuse ( LNM) = (LN)/(MN) Substitute the known values: (55^) = 4.25 cmMN Solve for MN: MN = 4.25 cm(55^) Calculate the value of (55^): (55^) ≈ 0.573576 Now, calculate MN: MN = 4.25 cm0.573576 MN ≈ 7.4099 cm Step 3: Round the length of MN to one decimal place. MN ≈ 7.4 cm The correct option is B. B. 7.4 cm 3 done, 2 left today. You're making progress.