In the triangle KMN, the angle is K = 75 degrees. angle n = 45 degrees. side KN = 4 root of 6. Find KM

By the condition of the problem, two angles of the triangle are known, we find the degree measure of the third angle:
KML angle = 180 ° – (75 ° + 45 °) = 180 ° – 120 ° = 60 °.
Let’s use the theorem of sines and write down the ratio of the sides to the sines of the opposite angles:
KN / sin M = KM / sin N;
KN / sin 60 ° = KM / sin 45 °;
4√6 / √3 / 2 = KM / √2 / 2.
We do not know the middle term of the proportion. We find it by dividing the products of the extreme terms by the known average term of the proportion:
KM = 4√6 * √2 / 2 / √3 / 2 = 4√6 * √2 / 2 * 2 / √3 = 8.
Answer: KM side = 8.