Question

В треугольнике АВС сторона AB=корень из 21 Радиус R описанной вокруг треугольника окружности равен Корень из 21 /2 Косинус угла САВ равен 2/корень из 7 Найдите: сторону ВС треугольника АВС; площадь треугольника АВС (в ответе запишите квадрат площади); отношение длины стороны АС к длине медианы, проведенной к этой стороне.

Answer 1

Решение задания приложено

Answer 2

Triangle ABC

In triangle ABC, we are given that side AB has a length of √21 and the radius R of the circumcircle is equal to √21/2. The cosine of angle CAV is equal to 2/√7.

To find the length of side BC, the area of triangle ABC, and the ratio of the length of side AC to the length of the median drawn to that side, we can use the following steps:

Finding the Length of Side BC

To find the length of side BC, we can use the Law of Cosines. The Law of Cosines states that in a 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 AB is given as √21, and angle CAV has a cosine of 2/√7. Let's substitute these values into the equation:

BC^2 = (√21)^2 + (√21)^2 - 2 * √21 * √21 * (2/√7)

Simplifying the equation:

BC^2 = 21 + 21 - 2 * 21 * (2/√7)

BC^2 = 42 - 84/√7

To simplify further, we can rationalize the denominator:

BC^2 = 42 - (84 * √7)/7

BC^2 = 42 - 12√7

Therefore, the length of side BC is √(42 - 12√7).

Finding the Area of Triangle ABC

To find the area of triangle ABC, we can use Heron's formula. Heron's formula states that for a triangle with sides a, b, and c, the area (A) can be calculated using the following formula:

A = √(s * (s - a) * (s - b) * (s - c))

where s is the semiperimeter of the triangle, given by:

s = (a + b + c) / 2

In our case, we know the lengths of sides AB (√21), BC (√(42 - 12√7)), and AC. We need to find the length of side AC.

Finding the Length of Side AC

To find the length of side AC, we can use the Law of Cosines again. We know that side AB has a length of √21, and the cosine of angle CAV is 2/√7. Let's substitute these values into the equation:

AC^2 = (√21)^2 + BC^2 - 2 * √21 * BC * (2/√7)

Simplifying the equation:

AC^2 = 21 + BC^2 - 4 * BC

AC^2 = 21 + (42 - 12√7) - 4 * √(42 - 12√7)

Therefore, the length of side AC is √(21 + (42 - 12√7) - 4 * √(42 - 12√7)).

Finding the Area of Triangle ABC (continued)

Now that we have the lengths of sides AB (√21), BC (√(42 - 12√7)), and AC (√(21 + (42 - 12√7) - 4 * √(42 - 12√7))), we can calculate the semiperimeter s:

s = (√21