Points M and N are selected on the sides BC and CD of the rectangle ABCD, respectively. So AMCN rhombus

Points M and N are selected on the sides BC and CD of the rectangle ABCD, respectively. So AMCN rhombus Find BC if the side of the rhombus is 18 and the angle ABD is 60 degrees.

1. Take the length of the segment BM as x.

2. By the condition of the problem, the quadrangle АМСN is a rhombus. All sides are equal.

Therefore, AM = MC = AM = 18 cm.

3. BC = (x + 18) cm.

4. Tangent ∠ABD tangent 60 ° = √3. AD / AV = √3. AD = BC since the opposite sides of the rectangle are equal.

Tangent ∠ABD = BC / AB = √3. AB = BC / √3 = (x + 18) / √3.

5. AB² + BM² = AM².

(x + 18) ² / 3 + x² = 18²;

(x² + 36x + 324) / 3 + x² = 324;

3x² + х² + 36x + 324 = 972;

4x² + 36x – 648 = 0;

x² + 9x – 162 = 0;

The first value x = (- 9 + √81 – 4 x 162) / 2 = (- 9 + 27) / 2 = 9 cm.

The second value x = (- 9 – 27) / 2 = – 18. Not accepted.

The length of the BM segment is 9 cm.

BC = x + 18 = 9 + 18 = 27 cm.

Answer: the length of the BC side of the rectangle is 27 cm.