In triangle LMN, LM = 23.7 m, MN = 16.6 m, and angle LMN = 106°. Calculate the distance LN, the area of triangle LMN, and the shortest distance from M to LN.

Here's the solution to the problem: a) Given: LM = 23.7 \, m MN = 16.6 \, m Angle LMN = 106^ i) Calculate the distance LN We can use the Cosine Rule to find the length of side LN. The Cosine Rule states c^2 = a^2 + b^2 - 2ab C. In our triangle LMN, we want to find LN (let's call it m), and we have sides LM (let's call it n) and MN (let's call it l) and the included angle LMN (angle M). LN^2 = LM^2 + MN^2 - 2(LM)(MN) (angle LMN) LN^2 = (23.7)^2 + (16.6)^2 - 2(23.7)(16.6) (106^) LN^2 = 561.69 + 275.56 - 2(393.42) (106^) LN^2 = 837.25 - 786.84 × (-0.275637...) LN^2 = 837.25 + 216.9010... LN^2 = 1054.1510... LN = sqrt(1054.1510...) LN ≈ 32.4676... Rounding to 3 significant figures: LN ≈ 32.5 \, m ii) Calculate the area of triangle LMN We can use the formula for the area of a triangle given two sides and the included angle: Area = (1)/(2)ab C. Area = (1)/(2) (LM)(MN) (angle LMN) Area = (1)/(2) (23.7)(16.6) (106^) Area = (1)/(2) (393.42) × 0.961261... Area = 196.71 × 0.961261... Area ≈ 189.100... Rounding to 3 significant figures: Area ≈ 189 \, m^2 iii) Calculate the shortest distance from M to LN. The shortest distance from point M to line LN is the perpendicular height (h) from M to LN. We can use the formula for the area of a triangle: Area = (1)/(2) × base × height. Here, LN is the base and h is the height. Area = (1)/(2) × LN × h We know the Area from part (ii) and LN from part (i). 189.100 = (1)/(2) × 32.4676 × h h = (2 × 189.100)/(32.4676) h = (378.200)/(32.4676) h ≈ 11.648... Rounding to 3 significant figures: h ≈ 11.6 \, m