The sides of the triangle are 9 and 24, and the angle between them is 60 degrees. Find the perimeter and area.

1. Vertices of the triangle A, B, C. AB = 9 units. BC = 24 units.

∠ABС = 60 °.

2. Calculate the length of the AC side using the cosine theorem:

AC² = AB² + BC² – 2 x AB x BC x cosine∠ABС.

Cosine 60 ° = 1/2.

AC² = 9² + 24² – 2 x 9 x 24 x 1/2 = 81 + 576 – 216 = 441.

AC = √441 = 21 units.

3. Calculate the perimeter of a given triangle (P):

P = 21 + 9 + 24 = 54 units.

4. Calculate the area (S) of a given triangle:

S = AB x BC x sine ∠ABС / 2.

Sine 60 ° = √3 / 2.

S = 9 x 24 x √3 / 2/2 = 54√3 units².

Answer: the area of ​​a given triangle is 54√3 units of measurement², the perimeter is 54 units of measurement.