One of the corners of an arbitrary triangle = 60 degrees opposite side = 7 cm the difference

One of the corners of an arbitrary triangle = 60 degrees opposite side = 7 cm the difference between the other two sides is 3 cm find these sides and the area of the triangle

Sides of a triangle:

a = 7 cm; b; c = b-3.

Cosine formula for the sides of a triangle:

a ^ 2 = b ^ 2 + c ^ 2 – 2cb * cosA, where A is the angle opposite side a.

cos60 ° = 1/2;

49 = b ^ 2 + (b-3) ^ 2 – 2 (b-3) b * cos 60 °;

49 = b ^ 2 + b ^ 2 – 6b + 9 – b ^ 2 + 3b;

40 = b ^ 2 – 3b;

b ^ 2 – 3b – 40 = 0;

Positive root:

b = (3 + √ (9 + 160)) / 2 = 8 cm.

c = 5;

Semi-perimeter p:

p = (a + b + c) / 2 = (7 + 8 + 5) / 2 = 10 cm;

Area according to Heron’s formula:

S = √p (p – a) (p – b) (p – c) = √ (10 (10 – 7) (10 – 8) (10 – 5)) = √ (10 * 3 * 2 * 5) = 10√3 cm ^ 2.

Answer: a = 7 cm, b = 8 cm, c = 5 cm, S = 10√3 cm ^ 2.