Question

Точки Р, К, L, M – середины сторон ромба АВСD. Докажите, что четырехугольник РКLM – прямоугольник.

Answer 1

Диагонали ромба пересекаются под прямым углом.

Так как стороны четырехугольника РКLM как средние линии треугольников, образованных сторонами ромба и диагоналями, параллельны диагоналям, они при пересечении образуют при вершинах четырехугольника прямые углы. Отсюда треугольник РКLM - прямоугольник. 

Answer 2

The question

The question asks to prove that the quadrilateral RKLM is a rectangle, given that points R, K, L, M are the midpoints of the sides of rhombus ABCD.

Facts about midpoints and rhombi

It is a known fact that in a rhombus, the line segments joining the midpoints of its sides are parallel to the sides of the rhombus. This is because the midpoints of the sides of any quadrilateral divide the sides into two congruent halves .

It is also a known fact that the diagonals of a rhombus are perpendicular bisectors of each other. The diagonals of a rhombus bisect each other at right angles .

Proving RKLM is a rectangle

Let's consider the rhombus ABCD. Points R, K, L, M are the midpoints of its sides AB, BC, CD, DA respectively.

By the first fact stated above, the line segments RK and LM are parallel to the sides AB and CD of the rhombus respectively. Similarly, the line segments RL and MK are parallel to the sides BC and DA respectively.

Also, since the diagonals AC and BD of the rhombus ABCD are perpendicular, the line segments RK and LM must intersect perpendicularly at their point of intersection, which is the center of the rhombus.

Therefore, by definition the quadrilateral RKLM is a rectangle.