Question

Abcd параллелограм ,угол a равен 30 ,ad равен 6см,m-середина bc,am пересекает bd в точке n ,cn пересекается с ab в точке p,ap = 6см ,найти площадь параллелограмма

Answer 1

BC=AD; BM=BC/2=8
треугольник APD подобен треугольнику MPB
AD/BM=AP/MP
16/8=6/MP
MP=3
из точек B и M проведем высоты BN и MK к стороне AD
AK^2+MK^2=AM^2
(AN+NK)^2+MK^2=(AP+PM)^2
MK=BN=ABsin(30)=AB/2
(AB*cos(30)+8)^2+(AB/2)^2=(6+3)^2 (решение ур-ния: AB=кореньиз (65)-4*кореньиз (3))
S(ABCD)=AB*AD*sin(30)=8*(кореньиз (65)-4*кореньиз (3))

Answer 2

Understanding the problem

You are given a parallelogram ABCD, where angle A is equal to 30 degrees and side AD is equal to 6 cm. The midpoint of side BC is M. The line segment AM intersects BD at point N, and line segment CN intersects AB at point P. It is also given that the length of segment AP is 6 cm. You are asked to find the area of the parallelogram.

Approach to solve the problem

To find the area of the parallelogram, we can use the formula: Area = base * height. In a parallelogram, any side can be considered as the base, and the corresponding height is the perpendicular distance between the base and the opposite side.

In this case, we have the length of side AD as 6 cm, and we need to find the corresponding height. To find the height, we can use the fact that the opposite sides of a parallelogram are parallel and equal in length. Therefore, we can consider side BC as the base and find the height as the perpendicular distance between BC and AD.

Once we find the height, we can calculate the area of the parallelogram using the formula mentioned above.

Finding the height of the parallelogram

To find the height of the parallelogram, we need to find the perpendicular distance between side BC and side AD. Let's denote this perpendicular distance as h.

Since m is the midpoint of BC, we can use the properties of a parallelogram to find the length of BM. Since BM is half the length of BC, we have BM = (1/2) * BC.

Now, we can use the fact that angle A is equal to 30 degrees to find the length of h. In triangle ABM, angle BMA is equal to 180 - angle A (parallelogram property). Therefore, angle BMA is equal to 150 degrees.

We can use the sine rule to find the length of h. The sine rule states that for any triangle ABC, (AB/sin(angle C)) = (BC/sin(angle A)) = (AC/sin(angle B)).

In triangle ABM, we have AB = 6 cm (given), angle BMA = 150 degrees, and angle BAM = 180 - angle BMA = 30 degrees.

Using the sine rule, we can write:

(AB/sin(angle BAM)) = (BM/sin(angle BMA))

Substituting the given values, we get:

(6/sin(30)) = ((1/2) * BC / sin(150))

Simplifying the equation, we have:

12 = (1/2) * BC / sin(150)

Solving for BC, we get:

BC = 12 * sin(150) * 2

Now that we have the length of BC, we can find the height h by using the formula:

h = BC * sin(angle A)

Substituting the given values, we get:

h = (12 * sin(150) * 2) * sin(30)

Calculating the area of the parallelogram

Now that we have the height h, we can calculate the area of the parallelogram using the formula:

Area = base * height

In this case, the base of the parallelogram is AD, which is given as 6 cm. Therefore, the area can be calculated as:

Area = 6 * h

Substituting the value of h, we get:

Area = 6 * (12 * sin(150) * 2) * sin(30)

Calculating this expression will give you the area of the parallelogram.