Question

В треугольнике АВС угол АВС=120 градусов, АВ=6, площадь равна 6 корней из трёх. Найти ВС.Напишите решение

Answer 1

Решение во вложенном файле

Answer 2

Problem Analysis

We are given a triangle ABC with angle ABC measuring 120 degrees, AB measuring 6 units, and the area of the triangle equal to 6 times the square root of 3. We need to find the length of side BC.

Solution

To find the length of side BC, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

In this case, side AB is the base of the triangle, and the height is the perpendicular distance from vertex C to side AB. Let's denote this perpendicular distance as h.

We can rewrite the formula for the area as:

Area = (1/2) * AB * h

Given that the area is equal to 6 times the square root of 3, we have:

6 * sqrt(3) = (1/2) * 6 * h

Simplifying the equation, we get:

sqrt(3) = h

So, the perpendicular distance from vertex C to side AB is equal to the square root of 3.

Now, we can use the Law of Cosines to find the length of side BC. The Law of Cosines states that for any triangle with sides a, b, and c, and angle C opposite side c, the following equation holds:

c^2 = a^2 + b^2 - 2ab * cos(C)

In our case, side BC is c, side AB is a, and angle ABC is C. We know that side AB is 6 units, and angle ABC is 120 degrees.

Substituting the values into the Law of Cosines equation, we get:

BC^2 = 6^2 + BC^2 - 2 * 6 * BC * cos(120)

Simplifying the equation, we have:

BC^2 = 36 + BC^2 + 12 * BC * cos(120)

Since cos(120) is equal to -1/2, we can further simplify the equation:

BC^2 = 36 + BC^2 - 12 * BC * (-1/2)

Simplifying again, we get:

BC^2 = 36 + BC^2 + 6 * BC

Rearranging the equation, we have:

0 = 36 + 6 * BC

Solving for BC, we find:

BC = -36 / 6 = -6

Since the length of a side cannot be negative, we discard this solution.

Therefore, there is no valid solution for the length of side BC in this case.

Conclusion

In the given triangle ABC with angle ABC measuring 120 degrees, AB measuring 6 units, and the area of the triangle equal to 6 times the square root of 3, there is no valid solution for the length of side BC.