Question

Докажите что что если диагонали параллелограмма взаимно перпендикулярны, то этот параллелограмм - ромб

Answer 1

Proof that if the diagonals of a parallelogram are mutually perpendicular, then the parallelogram is a rhombus

To prove that if the diagonals of a parallelogram are mutually perpendicular, then the parallelogram is a rhombus, we can use the following steps:

1. Start by assuming that we have a parallelogram ABCD, where AB is parallel to CD and AD is parallel to BC.

2. Let the diagonals of the parallelogram intersect at point O.

3. We need to prove that if the diagonals AC and BD are mutually perpendicular, then the parallelogram ABCD is a rhombus.

4. To prove this, we can use the properties of a parallelogram and the fact that the diagonals are mutually perpendicular.

5. Property 1: In a parallelogram, opposite sides are equal in length. Therefore, AB = CD and AD = BC.

6. Property 2: In a parallelogram, opposite angles are equal. Therefore, angle A = angle C and angle B = angle D.

7. Property 3: In a parallelogram, the diagonals bisect each other. Therefore, AO = OC and BO = OD.

8. Property 4: In a quadrilateral, the sum of the interior angles is 360 degrees. Therefore, angle A + angle D = 180 degrees.

9. Now, let's consider the diagonals AC and BD.

10. Since AC and BD are mutually perpendicular, we can conclude that angle AOB = angle COD = 90 degrees.

11. From Property 2, we know that angle A = angle C and angle B = angle D.

12. Therefore, angle AOB = angle COD implies that angle A = angle C = 90 degrees and angle B = angle D = 90 degrees.

13. This means that all angles of the parallelogram ABCD are right angles, making it a rectangle.

14. Additionally, since AB = CD and AD = BC (Property 1), and opposite sides are equal in a rectangle, we can conclude that AB = BC = CD = AD.

15. Finally, a rectangle with equal sides is a rhombus.

16. Therefore, if the diagonals of a parallelogram are mutually perpendicular, then the parallelogram is a rhombus.

In conclusion, if the diagonals of a parallelogram are mutually perpendicular, then the parallelogram is a rhombus.