In a triangle, the two sides are 10 and 12 cm, and the angle between them is 45 degrees. Find the area of the triangle.

By hypothesis, △ ABC is given. Consider △ ABC: AB = 10 cm and AC = 12 cm – sides △ ABC, ∠A = 45 ° – angle between sides AB and AC.

The area of a triangle can be found through the length of its two sides and the angle between these sides by the formula:

S = (a * b * sin α) / 2,

where a and b are the lengths of the two sides of the triangle, α is the angle between sides a and b.

Substitute the value condition data into the area formula and find it:

S = (AB * AC * sin∠A) / 2 = (10 * 12 * sin45 °) / 2 = (120 * √2 / 2) / 2 = (120 * √2) / 2: 2 = 60√2 : 2 = (60√2) / 2 = 30√2 (cm²).

Answer: S = 30√2 cm².