Question

В треугольнике АВС АС=ВС АН высота тангенс ВАС = 2.Найдите тангенс ВАН?

Answer 1

треугольник равнобедренный => углы при основании равны CBA=CAB

из треуг.AHB tg(BAH) = tg(90-HBA) = ctg(HBA) = ctg(CBA) = ctg(CAB) = 1/tg(CAB) = 1/2

 

Answer 2

Task: Find the tangent of angle VAN in triangle ABC, given that AC = BC and AN is the height of the triangle, and the tangent of angle VAC is 2.

To find the tangent of angle VAN, we need to use the information given in the problem. Let's break down the problem and solve it step by step.

Given information:

- Triangle ABC - AC = BC - AN is the height of the triangle - The tangent of angle VAC is 2

Solution:

1. Let's label the triangle ABC and the angles as shown below:

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

2. Since AC = BC, we can conclude that triangle ABC is an isosceles triangle.

3. Let's label the height AN as shown below:

       A
      / \
     /   \
    /     \
   /       \
  /         \
 B-----------C
     |       |
     |       |
     |       |
     N       |

4. We are given that the tangent of angle VAC is 2. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. In this case, the opposite side is AN and the adjacent side is AC.

5. Using the given information, we can write the equation:

tangent(VAC) = AN / AC = 2

6. Since AC = BC, we can substitute AC with BC in the equation:

tangent(VAC) = AN / BC = 2

7. Now, we need to find the tangent of angle VAN. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. In this case, the opposite side is AN and the adjacent side is VN.

8. To find the tangent of angle VAN, we need to find the length of VN. Since triangle ABC is an isosceles triangle, we can conclude that VN is equal to half of BC.

9. Let's substitute VN with BC/2 in the equation:

tangent(VAN) = AN / (BC/2)

10. Since AC = BC, we can substitute BC with AC in the equation:

tangent(VAN) = AN / (AC/2)

11. Simplifying the equation, we get:

tangent(VAN) = 2 * AN / AC

12. We know that the tangent of angle VAC is 2, so we can substitute the value of AN / AC with 2:

tangent(VAN) = 2 * 2 = 4

Answer:

The tangent of angle VAN is 4.

Please let me know if I can help you with anything else.