Question

в параллелограмме abcd точка m лежит на стороне ab так что bm = am найдите площадь bcm если площадь параллелограмма равна 32

Answer 1

Постройте рисунок, будет понятней.

Поставим на стороне cd некую точку n так, чтобы она тоже лежала посередине: cn=cd

Тогда параллелограмм разделится на две равных половинки mbcn и amnd, каждая тоже параллелограмм и площадью по 16.

В параллелограмме mbcn ищется площадь треугольника, построенного на двух сторонах и дианогали. Диагональ всегда делит параллелограмм на две равновеликих части - значит площадь треугольника будет равна половине площади маленького параллелограмма и равна 8

Answer 2

Problem Analysis

We are given a parallelogram ABCD, where point M lies on side AB such that BM = AM. We need to find the area of triangle BCM, given that the area of the parallelogram ABCD is 32.

Solution

To find the area of triangle BCM, we need to know the lengths of its base and height. Let's denote the length of BM as x.

Since BM = AM, we can conclude that the length of AB is 2x.

Now, let's consider the area of the parallelogram ABCD. The area of a parallelogram is given by the product of its base and height. In this case, the base is AB and the height is the perpendicular distance between AB and CD.

Since ABCD is a parallelogram, the perpendicular distance between AB and CD is equal to the perpendicular distance between AD and BC. Let's denote this distance as h.

Therefore, the area of ABCD is given by AB * h = 2x * h.

We are given that the area of ABCD is 32, so we have the equation 2x * h = 32.

Now, let's consider triangle BCM. The base of triangle BCM is BC, which is equal to CD (since ABCD is a parallelogram). Let's denote the length of BC (or CD) as y.

The height of triangle BCM is h, which we already know from the equation 2x * h = 32.

Therefore, the area of triangle BCM is given by (1/2) * BC * h = (1/2) * y * h.

To find the area of triangle BCM, we need to find the value of y.

From the given information, we know that BM = AM, so the length of AB is 2x. Since ABCD is a parallelogram, BC is equal to CD, so BC = y.

We can use these relationships to find the value of y.

Calculation

From the equation 2x * h = 32, we can solve for h: h = 32 / (2x) h = 16 / x

Since ABCD is a parallelogram, the opposite sides are equal in length. Therefore, BC = CD = y.

We also know that BM = AM, so the length of AB is 2x.

Using these relationships, we can find the value of y: 2x = BC = CD = y

Now, we can substitute the value of h and y into the formula for the area of triangle BCM: Area of BCM = (1/2) * BC * h Area of BCM = (1/2) * y * (16 / x) Area of BCM = 8y / x

Therefore, the area of triangle BCM is 8y / x.

Answer

The area of triangle BCM is 8y / x.

Note: To find the specific values of y and x, we would need additional information or measurements from the problem statement.