Question

Биссектриса углов А и Д паралелограма АВСД пересекаются в точке М лежащей на стороне ВС луч ДМ пересекает прямую АБ в точке Н . Найдите периметр АВСД если АН=10 см

Answer 1

Сумма углов параллелограмма прилежащих к одной стороне=180 град
.уг.А+уг Д=180  Они разделены биссектрисами попалам,значит в трке АМД сумма двух острых углов =90 гр., следовательно  уголАМД  тоже =90 гр.
Продлим Отрезок АМ тоже до пересечения с продолжением стороныДС  в точке К. соединим точки Н и К Рассотрим четырехугольникАНКД. Он состоит  из 4 равных прямоугольников , где стороны взаимно перпендикулярны и гипотенузы этих прямоугольников образуют ромб . В ромбе все стороны =10 э   периметр параллелограмма АВСД это сумма 2 сторон по 10 + 2 стороны по 5  Р=30

Answer 2

Bisector of Angles A and D in Parallelogram ABCD

In the given parallelogram ABCD, the bisector of angles A and D intersect at point M, which lies on side BC. The ray DM intersects line AB at point N. We need to find the perimeter of ABCD given that AN = 10 cm.

To find the perimeter, we need to determine the lengths of the sides of the parallelogram. Let's break down the problem step by step.

Step 1: Finding the Length of Side AB

Since DM is a ray that intersects line AB at point N, we can conclude that AN is equal to the length of segment AB. Therefore, AN = AB = 10 cm.

Step 2: Finding the Length of Side BC

Since M lies on side BC, we need to find the length of segment BM. To do this, we can use the fact that the bisector of angles A and D passes through point M. This means that the lengths of segments AM and DM are equal.

Let's assume that the length of segment AM is x cm. Then, the length of segment DM is also x cm. Since AN = AB = 10 cm, we can write the following equation:

AM + BM = AB x + BM = 10

Now, let's consider triangle ABM. In this triangle, we have two sides (AM and AB) and the included angle (angle BAM). We can use the Law of Cosines to find the length of side BM.

The Law of Cosines states that in a triangle with sides a, b, and c, and the included angle C, the following equation holds:

c^2 = a^2 + b^2 - 2ab * cos(C)

In triangle ABM, we have: - Side a: AM (length x) - Side b: AB (length 10 cm) - Included angle C: angle BAM (which is half of angle A)

Using the Law of Cosines, we can write the equation for side BM:

BM^2 = x^2 + 10^2 - 2 * x * 10 * cos(angle BAM)

Since the bisector of angle A passes through point M, angle BAM is half of angle A. Therefore, we can write:

BM^2 = x^2 + 100 - 20x * cos(angle A/2)

To find the value of BM, we need to know the value of angle A. Unfortunately, the given information does not provide the value of angle A. Without this information, we cannot determine the exact length of side BC.

Step 3: Finding the Length of Side CD

Since ABCD is a parallelogram, opposite sides are equal in length. Therefore, CD is equal to AB, which is 10 cm.

Step 4: Finding the Length of Side DA

Since ABCD is a parallelogram, opposite sides are equal in length. Therefore, DA is equal to BC. However, we were unable to determine the exact length of side BC in Step 2 due to the lack of information about angle A.

Step 5: Calculating the Perimeter

To calculate the perimeter of ABCD, we need to add up the lengths of all four sides:

Perimeter = AB + BC + CD + DA

Since we know that AB = 10 cm and CD = 10 cm, we can write:

Perimeter = 10 + BC + 10

However, without the exact length of side BC, we cannot calculate the perimeter of ABCD.

In conclusion, without the value of angle A or the exact length of side BC, we cannot determine the perimeter of parallelogram ABCD.