In the MKR triangle it is known that KP = 8cm, angle K = 106 degrees, angle P = 32 degrees. Find the unknown side of MP.

The triangle sum theorem states that the sum of all interior angles of any triangle is 180 °. So in △ MKP:

∠M + ∠K + ∠P = 180 °;

∠M + 106 ° + 32 ° = 180 °;

∠M = 180 ° – 138 °;

∠M = 42 °.

By the sine theorem:

KP / sin∠M = MP / sin∠K.

Substitute the known values:

8 / sin42 ° = x / sin106 °;

x = (8 * sin106 °) / sin42 ° (proportional).

According to the reduction formulas:

sin106 ° = sin (180 ° – 74 °) = sin74 °.

Thus:

x = (8 * sin74 °) / sin42 °.

Using the Bradis tables, we find the values of the sines:

sin74 ° = 0.9613;

sin42 ° = 0.6691.

Let’s find the length of the side MP:

MP = (8 * 0.9613) / 0.6691 = 11.4936482 ≈ 11.5 (cm).

Answer: MP = 11.5 cm.