Question

Докажите что если в параллелограмме диагонали равны то параллелограм является прямоугольником

Answer 1

Параллелограмм не плямоугольник, ведь у него нет прямых углов

Answer 2

Пусть в параллелограмме АВСD диагонали АС и DСА равны. Треугольники АВD и DCA  равны по трем сторонам (АВ=DC, BD=CA, AD- общая сторона). Отсюда следует, что угол А равен углу D. Так как в параллелограмме противоположные углы равны, то угол А равен углу С и угол В равен углу D. Параллелограмм- выпуклый четырехугольник, поэтому углы А+В+С+D=360 градусов. Следовательно, угол А = углу В = углу С = углу D =90 градусов т. е. параллелограмм АВСD является прямоугольником

Answer 3

Checking if Diagonals of a Parallelogram are Equal

To prove that if the diagonals of a parallelogram are equal, then the parallelogram is a rectangle, we can use the properties of parallelograms and rectangles.

Parallelogram Properties: 1. Opposite sides of a parallelogram are equal in length. 2. Opposite angles of a parallelogram are equal in measure. 3. The diagonals of a parallelogram bisect each other.

Rectangle Properties: 1. A rectangle is a parallelogram with four right angles. 2. The diagonals of a rectangle are equal in length.

Proof

Let's assume we have a parallelogram ABCD with diagonals AC and BD. If the diagonals are equal, i.e., AC = BD, we can prove that the parallelogram is a rectangle.

Step 1: Opposite Sides Since ABCD is a parallelogram, opposite sides are equal. Let's assume AB = CD and AD = BC.

Step 2: Diagonals Bisect Each Other The diagonals of a parallelogram bisect each other. This means that the line segments created by the intersection of the diagonals are equal. Let's assume that the point of intersection is E, so AE = EC and BE = ED.

Step 3: Applying the Pythagorean Theorem In a rectangle, the diagonals are equal in length. Using the Pythagorean theorem, we can show that if the diagonals are equal, then the parallelogram is a rectangle.

By applying the Pythagorean theorem to triangles ABE and CDE, we get: - AE^2 + BE^2 = AB^2 - CE^2 + DE^2 = CD^2

Since AB = CD and AE = EC, and BE = ED, we can substitute these values into the equations.

If AE = EC and BE = ED, then: - AE^2 + BE^2 = CE^2 + DE^2

This implies that AB^2 = CD^2, which means that the opposite sides of the parallelogram are equal and the angles are right angles, making it a rectangle.

Therefore, if the diagonals of a parallelogram are equal, then the parallelogram is a rectangle.

Conclusion: If the diagonals of a parallelogram are equal, then the parallelogram is a rectangle.