Question

В параллелограмме АВСД проведена прямая МК (МК и АВ не параллельны, МϵВС и КϵАД) пересекающая диагональ ВД в точке Р. Найти площадь параллелограмма АВСД, если площади треугольников ВРМ ,КРД и АВК соответственно равны 4 , 9 и 22,5 .

Answer 1

 Из подобия тех же треугольников следует что                             
  $\frac{DP}{PB} = \frac{3}{2} \\ \frac{KP}{PM} = \frac{3}{2} \\ \\ DP=3y\\ PB=2y\\ KP=3x\\ PM= 2x\\ \\ \frac{h_{KPD}}{h_{PBM}} = \frac{3z}{2z} \\ AK*5z=45 \\ KD*3z = 18 \\ \frac{AK}{KD} = \frac{3}{2} \\ S_{ABCD} = \frac{5KD}{2}*5z = \frac{25*KD*z}{2} = 25*\frac{6}{2} = 75$

Answer 2

Problem Analysis

We are given a parallelogram ABCD with a diagonal BD. A line MK is drawn through the parallelogram, intersecting the diagonal BD at point P. The areas of triangles VRM, KRD, and AVK are given as 4, 9, and 22.5 respectively. We need to find the area of the parallelogram ABCD.

Solution

To find the area of the parallelogram ABCD, we can use the fact that the area of a parallelogram is equal to the product of its base and height.

Let's denote the base of the parallelogram as b and the height as h. We need to find the values of b and h.

Since MK and AB are not parallel, we can see that triangles VRM and KRD are similar. This means that their corresponding sides are proportional. Let's denote the length of VR as x.

From the given information, we know that the area of triangle VRM is 4. Using the formula for the area of a triangle (area = 0.5 * base * height), we can write:

0.5 * x * h = 4

Similarly, the area of triangle KRD is 9, so we have:

0.5 * x * h = 9

Dividing the second equation by the first equation, we get:

(0.5 * x * h) / (0.5 * x * h) = 9 / 4

Simplifying, we find:

h = 9 / 4

Now, let's consider triangle AVK. The area of this triangle is given as 22.5. Using the formula for the area of a triangle, we can write:

0.5 * b * h = 22.5

Substituting the value of h we found earlier, we get:

0.5 * b * (9 / 4) = 22.5

Simplifying, we find:

b = (22.5 * 4) / 9

Now that we have the values of b and h, we can calculate the area of the parallelogram ABCD:

Area = b * h

Substituting the values we found, we get:

Area = ((22.5 * 4) / 9) * (9 / 4)

Simplifying, we find:

Area = 22.5

Therefore, the area of the parallelogram ABCD is 22.5.

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