Question

ВН и ВК - высоты пароллелограмма АВСD. Угол между ними 45(градусов).Высота ВН делит сторону АD на отрезки АН = 8см., НD= 4см. Найдите площаль параллелограмма.

Answer 1

Finding the Area of a Parallelogram

To find the area of a parallelogram, we can use the formula: Area = base × height. In this case, the base of the parallelogram is the length of side AD, and the height is the length of the segment VN.

Given that segment VN divides side AD into two segments, AN and ND, with lengths AN = 8 cm and ND = 4 cm, respectively, we can calculate the length of side AD by adding the lengths of AN and ND: AD = AN + ND = 8 cm + 4 cm = 12 cm.

Now, we need to find the length of segment VN, which is the height of the parallelogram. To do this, we can use the fact that VN is perpendicular to AD and forms a right triangle with AN and ND. Since the angle between VN and AD is given as 45 degrees, we have a right triangle with a 45-degree angle.

Using trigonometry, we can find the length of VN. In a right triangle with a 45-degree angle, the lengths of the two legs are equal. Therefore, VN is equal to the length of either AN or ND, which is 8 cm.

Now that we have the base (AD = 12 cm) and the height (VN = 8 cm), we can calculate the area of the parallelogram using the formula mentioned earlier:

Area = base × height = AD × VN = 12 cm × 8 cm = 96 cm².

Therefore, the area of the parallelogram is 96 square centimeters.