Area of triangle MNP equals 14; angle M equals 45 MP equals 7 find side NP.

Let us write the formula for the area of a triangle through two sides and the sine of the angle between them and with its help we calculate the side MN.
S = 1/2 * MN * MP * sin ∠M.
MN = 2S / (MP * sin ∠M) = 2 * 14/7 * 1 / √2 = 4√2.
We know two sides of the triangle, MN and MP, find the third NP using the cosine theorem in an arbitrary triangle:
a2 = b2 + c2 – 2 * b * c * cosγ;
NP² = MN² + MP² – 2 * MN * MP * cos ∠M = 32 + 49 – 2 * 4√2 * 7 * 1 / √2 = 25;
NP = √25 = 5.
Answer: NP side is 5.