Question

АВСД – параллелограмм. АМ и ДН – биссектрисы углов ВАД и АДС. Точки М и Н делят сторону ВС на три части, где МН =8см; Р АВСД = 44 см. Найти стороны параллелограмма.

Answer 1

АВМ-равнобедренный треугольник, тк угол BMA=MAD(накрестлеж)=ВАМ. Значит ВА=ВМ. BCD- тоже равнобедренный, углы CND=NDA=CDN. CD=NC.MC=AB-8. По условию AB+BC=22(половина Р). Тогда AB+BM+MC=22; AB+AB+AB-8=22; 3AB=30;AB=10. BC=12

Answer 2

Parallelogram ABCD and Bisectors AM and DN

In the given problem, we have a parallelogram ABCD, where AM and DN are the bisectors of angles BAD and ADS, respectively. Points M and N divide side BC into three parts, where MN = 8 cm, and the perimeter of parallelogram ABCD is 44 cm. We need to find the lengths of the sides of the parallelogram.

To solve this problem, we can use the properties of parallelograms and the given information.

Properties of Parallelograms

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.

Solution

Let's denote the length of side AB as a and the length of side AD as b.

Since ABCD is a parallelogram, we know that AB = CD and AD = BC.

Using the property of opposite sides being equal, we can write the following equations:

AB = CD = a -- (1) AD = BC = b -- (2)

We also know that MN divides side BC into three parts, where MN = 8 cm. This means that BN = 2MN = 16 cm and NC = MN = 8 cm.

Using the property of opposite sides being equal, we can write the following equations:

BN = CD = a -- (3) NC = AD = b -- (4)

The perimeter of parallelogram ABCD is given as 44 cm. The perimeter of a parallelogram is the sum of the lengths of all four sides. Using equations (1), (2), (3), and (4), we can write the following equation:

2a + 2b = 44

Simplifying the equation, we get:

a + b = 22 -- (5)

Now, we have two equations (1) and (5) with two unknowns (a and b). We can solve these equations simultaneously to find the values of a and b.

From equation (1), we have:

a = CD = AB

Substituting this value in equation (5), we get:

AB + b = 22

Simplifying the equation, we get:

AB = 22 - b -- (6)

Now, we have the value of AB in terms of b. To find the value of b, we can substitute this expression in equation (3):

BN = a = 22 - b

Substituting this value in equation (3), we get:

22 - b = 16

Solving for b, we find:

b = 22 - 16 = 6

Substituting the value of b in equation (6), we can find the value of AB:

AB = 22 - 6 = 16

Therefore, the lengths of the sides of the parallelogram are AB = 16 cm and AD = BC = 6 cm.

Answer

The lengths of the sides of the parallelogram are AB = 16 cm and AD = BC = 6 cm.