Question

В параллелограмме АМРК диагональ МК перпендикулярна стороне АМ, диагонали пересекаются в точке О. Найдите сторону МА, если иагонали МК и АР равны 18 и 82 соответственно

Answer 1

 таак как диаогональ точкой пересечения делаться пополам, то МО=9,а ОА=41. Следовательно МА= \sqrt{1681-81}= 40

Answer 2

в параллелограмме диагонали точкой пересечения делятся пополам

МО=9

АО=41

получаем прямоугольный треугольник АОМ 

по Т Пифагора: АМ=корень(АО^2-MO^2)

AM=корень(1681-81)=40 

Answer 3

Finding the Length of Side MA in Parallelogram AMRK

To find the length of side MA in parallelogram AMRK, we can use the properties of diagonals in a parallelogram. Given that diagonal MK is perpendicular to side AM and the diagonals intersect at point O, and the lengths of diagonals MK and AR are 18 and 82 respectively, we can calculate the length of side MA.

Calculation

In a parallelogram, the diagonals bisect each other. This means that each diagonal is divided into two equal parts by the point of intersection.

Let's denote the length of MO as x. Since the diagonals bisect each other, the length of AO will also be x.

Using the given information, we can set up the following equations: 1. \( MO = x \) 2. \( AO = x \) 3. \( AR = 82 \) 4. \( MK = 18 \)

We can use the Pythagorean theorem to find the length of side MA. According to the theorem, in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

So, we have: \[ MA^2 = MO^2 + AO^2 \]

Substituting the values we have: \[ MA^2 = x^2 + x^2 \] \[ MA^2 = 2x^2 \]

Now, we need to find the value of x. To do this, we can use the fact that the diagonals bisect each other, so the sum of the squares of the lengths of the diagonals is equal to the sum of the squares of the lengths of the four segments formed by the intersection of the diagonals.

We can use the following equation: \[ MK^2 + AR^2 = 2(MO^2 + AO^2) \]

Substituting the given values: \[ 18^2 + 82^2 = 2(x^2 + x^2) \] \[ 324 + 6724 = 4x^2 \] \[ 7048 = 4x^2 \] \[ x^2 = 1762 \] \[ x = \sqrt{1762} \]

Now that we have the value of x, we can find the length of side MA: \[ MA = \sqrt{2x^2} \] \[ MA = \sqrt{2 \times 1762} \] \[ MA = \sqrt{3524} \] \[ MA ≈ 59.38 \]

Answer

So, the length of side MA in parallelogram AMRK is approximately 59.38.