A rhombus is inscribed in a right-angled triangle with an acute angle of 60 degrees. The vertex of this angle

| September 3, 2021 | education

A rhombus is inscribed in a right-angled triangle with an acute angle of 60 degrees. The vertex of this angle is common, and the other 3 vertices of the rhombus lie on the sides of the triangle. Find the lengths of the sides of the triangle if it is known that the length of the side of the rhombus is 12 cm.

Let this triangle have sides a, b and c. A rhombus cuts off a similar triangle from a triangle with a hypotenuse c1 = 12 and a leg b1 = 1/2 * c1 = 1/2 * 12 cm = 6 cm. (Like a leg opposite an angle of 30 °)

Since the side of the rhombus is 12 cm on the leg in, then b = 12 cm + b1 = 12 cm + 6 cm = 18 cm.But this leg in lies opposite an angle of 30 °, then the hypotenuse c = 2 * b = 2 * 18 cm = 36 cm.

Determine the leg a: a = √ (c ^ 2 – b ^ 2) = √ (36 ^ 2 – 18 ^ 2) = √ (1296 – 324) = √972 = √ (81 * 4 * 3) = 9 * 2 * √3 = 18 * √3 (cm).

Answer: c = 36 cm; h = 18 cm; a = 18√3 cm.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.