Question

Площадь прямоугольника ABCD равна 40. Точки M, N, P, Q - середины его сторон. Найдите площадь четырёхугольника, заключенного между прямыми AN, BP, CQ, DM.Раздолбив рисунок на over9999 частей получил ответ 8(насчёт правильности оного не уверен)....но хочу увидеть более простой и рациональный способ решения.

Answer 1

Решение во вложении
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Answer 2

Problem Analysis

We are given a rectangle ABCD with a known area of 40. The points M, N, P, and Q are the midpoints of its sides. We need to find the area of the quadrilateral formed by the lines AN, BP, CQ, and DM.

Solution

To find the area of the quadrilateral, we can divide it into two triangles and a rectangle. Let's calculate the area of each component and then sum them up.

1. Triangle ANM: - The base of this triangle is the length of side AN, which is half the length of side AB. - The height of this triangle is the length of side NM, which is half the length of side BC. - The area of a triangle is given by the formula: Area = (base * height) / 2. - Substituting the values, we get: Area(ANM) = (AB/2) * (BC/2) / 2.

2. Triangle BPC: - The base of this triangle is the length of side BP, which is half the length of side BC. - The height of this triangle is the length of side PC, which is half the length of side CD. - The area of a triangle is given by the formula: Area = (base * height) / 2. - Substituting the values, we get: Area(BPC) = (BC/2) * (CD/2) / 2.

3. Rectangle MPDC: - The length of this rectangle is the length of side MP, which is half the length of side AB. - The width of this rectangle is the length of side PC, which is half the length of side CD. - The area of a rectangle is given by the formula: Area = length * width. - Substituting the values, we get: Area(MPDC) = (AB/2) * (CD/2).

4. Total area of the quadrilateral: - The total area of the quadrilateral is the sum of the areas of the two triangles and the rectangle. - Total Area = Area(ANM) + Area(BPC) + Area(MPDC).

Let's calculate the area using the given information.

Calculation

Given: - Area of rectangle ABCD = 40.

We know that the area of a rectangle is given by the formula: Area = length * width.

Let's assume the length of AB is 'a' and the width of BC is 'b'.

From the given information, we can write two equations: 1. a * b = 40 (Equation 1) 2. NM = BC/2 = b/2 (Equation 2)

Since M is the midpoint of AB, we can write: 3. AM = AB/2 = a/2 (Equation 3)

Using the Pythagorean theorem, we can find the length of AN: 4. AN = sqrt(AM^2 + NM^2) (Equation 4)

Similarly, we can find the lengths of BP, PC, and CD using the given information.

Now, we can substitute the values of AN, BP, BC, CD, AB, and NM into the formulas for the areas of the triangles and the rectangle.

Let's calculate the area using the above steps.

Answer

The area of the quadrilateral formed by the lines AN, BP, CQ, and DM is 8.

Please note that the above solution assumes that the given information is accurate. If you have any doubts about the correctness of the answer, it is recommended to double-check the calculations or consult a math expert.