Question

В ромбе ABCD на стороне BC отмечена точка К такая, что КС: ВК = 3:1.Найдите площадь ABK,если площадь ромба равна 48см^2 Желательно с рисунком, но можно и без

Answer 1

Решение смотри на фотографии

Answer 2

Finding the Area of Triangle ABK in Rhombus ABCD

To find the area of triangle ABK in rhombus ABCD, we need to use the given information that the ratio of KS to VK is 3:1.

Let's proceed with the solution:

1. Draw a diagram of rhombus ABCD. Label the points A, B, C, and D as shown in the diagram.

         A
        / \
       /   \
      /     \
     /       \
    B---------C
     \       /
      \     /
       \   /
        \ /
         D
   

2. Mark point K on side BC such that the ratio of KS to VK is 3:1. This means that KS is three times VK.

3. Let's assume that VK is x cm. Therefore, KS is 3x cm.

4. Since ABCD is a rhombus, the diagonals AC and BD bisect each other at right angles. Let's label the point of intersection of the diagonals as O.

5. Now, we can see that triangle ABK is a right triangle with right angle BAK. The hypotenuse of this triangle is AB, which is also a side of the rhombus.

6. Since the diagonals of a rhombus bisect each other at right angles, triangle AOB is also a right triangle with right angle ABO. The hypotenuse of this triangle is AO, which is half the length of the diagonal AC.

7. Let's assume that the length of the diagonal AC is d cm. Therefore, the length of AO is d/2 cm.

8. Since triangle AOB is a right triangle, we can use the Pythagorean theorem to find the length of AB in terms of d.

   AB^2 = AO^2 + OB^2
   AB^2 = (d/2)^2 + (d/2)^2
   AB^2 = (d^2/4) + (d^2/4)
   AB^2 = (2d^2/4)
   AB^2 = (d^2/2)
   AB = sqrt(d^2/2)
   AB = d/sqrt(2)
   

9. Now, we can find the area of triangle ABK using the formula for the area of a right triangle: area = (base * height) / 2.

The base of triangle ABK is AB, which is d/sqrt(2) cm, and the height of triangle ABK is VK, which is x cm.

Therefore, the area of triangle ABK is:

   Area = (AB * VK) / 2
        = (d/sqrt(2) * x) / 2
        = (dx) / (2 * sqrt(2))
   

10. We are given that the area of the rhombus ABCD is 48 cm^2. The area of a rhombus can be calculated using the formula: area = (diagonal1 * diagonal2) / 2.

Since ABCD is a rhombus, the diagonals AC and BD are equal in length. Let's assume that the length of the diagonals is d cm.

Therefore, the area of the rhombus ABCD is:

    Area = (AC * BD) / 2
         = (d * d) / 2
         = d^2 / 2
    

11. We are given that the area of the rhombus ABCD is 48 cm^2. Therefore, we can write the equation:

    d^2 / 2 = 48
    d^2 = 48 * 2
    d^2 = 96
    d = sqrt(96)
    

12. Now that we have found the length of the diagonal AC, we can substitute it into the formula for the area of triangle ABK:

    Area = (dx) / (2 * sqrt(2))
         = (sqrt(96) * x) / (2 * sqrt(2))
         = (sqrt(96) * x) / (2 * sqrt(2))
         = (sqrt(96) * x) / (2 * sqrt(2))
         = (4 * sqrt(6) * x) / (2 * sqrt(2))
         = (2 * sqrt(6) * x) / sqrt(2)
         = sqrt(6) * x
    

13. We know that the ratio of KS to VK is 3:1. Therefore, VK is one-fourth of the length of BC.

Since BC is equal to the length of the diagonal AC, which is sqrt(96) cm, VK is:

    VK = (1/4) * sqrt(96)
       = sqrt(96) / 4
       = sqrt(16 * 6) / 4
       = (4 * sqrt(6)) / 4
       = sqrt(6)
    

14. Finally, we can substitute the value of VK into the formula for the area of triangle ABK:

    Area = sqrt(6) * x
         = sqrt(6) * sqrt(6)
         = 6 cm^2
    

Therefore, the area of triangle ABK is 6 cm^2.

I hope this helps! Let me know if you have any further questions.