Question

В ромбе MTHD точка K принадлежит диагонали TD. TK:KD=2:1. Точка E делит отрезок HD пополам. Используя метод координат, докажите, что точка K принадлежит отрезку ME и делит его в отношении 1:2

Answer 1

Будет 1,5 наверно так

Answer 2

Given Information

The problem provides the following information: - In rhombus MTHD, point K lies on diagonal TD, and TK:KD=2:1. - Point E divides segment HD in half.

Method of Coordinates

To prove that point K belongs to segment ME and divides it in the ratio 1:2, we can use the method of coordinates. Let's assume the coordinates of points M, T, H, D, and E as follows: - M(x1, y1) - T(x2, y2) - H(x3, y3) - D(x4, y4) - E(x5, y5)

Proof

1. First, let's find the coordinates of point K using the given information that TK:KD=2:1. We can use the section formula to find the coordinates of K. - Let's assume the coordinates of point K as (x, y). - Using the section formula, the coordinates of K can be calculated as: ``` x = (2*x4 + x2) / 3 y = (2*y4 + y2) / 3 ```

2. Next, we'll find the coordinates of point E, which divides segment HD in half. We can use the midpoint formula to find the coordinates of E. - The coordinates of E can be calculated as: ``` x5 = (x3 + x4) / 2 y5 = (y3 + y4) / 2 ```

3. Now, we need to find the coordinates of point M, which is the midpoint of segment TD. Using the midpoint formula, the coordinates of M can be calculated as: ``` x1 = (x2 + x4) / 2 y1 = (y2 + y4) / 2 ```

4. Finally, we'll use the distance formula to verify if point K lies on segment ME and divides it in the ratio 1:2. - The distance between points M and E is given by: ``` ME = sqrt((x5 - x1)^2 + (y5 - y1)^2) ``` - The distance between points M and K is given by: ``` MK = sqrt((x - x1)^2 + (y - y1)^2) ``` - The distance between points K and E is given by: ``` KE = sqrt((x5 - x)^2 + (y5 - y)^2) ```

5. By calculating the ratios ME:MK and MK:KE, we can verify if point K belongs to segment ME and divides it in the ratio 1:2.

Conclusion

Using the method of coordinates, we can calculate the coordinates of points K, E, and M, and then verify if point K lies on segment ME and divides it in the ratio 1:2.