Question

Дано:∆АВСAB=2BC=4 AC=2√3 Найти: угол А (Напишите решение) ​

Answer 1

Ответ:

∠A=90°

Объяснение:

Теорема косинусів:

cos∠A=(AB²+AC²-BC²)/(2*AB*AC)=

=(2²+(2√3)²-4²)/(2*2*2√3)=

=(4+12-16)/(8√3)=0

∠A=arccos0=90°

Answer 2

Given Information:

We are given that triangle ABC has the following properties: - AB = 2BC - AC = 2√3

Solution:

To find angle A, 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, we want to find angle A, so we can rewrite the equation as follows:

AC^2 = AB^2 + BC^2 - 2AB * BC * cos(A)

Substituting the given values, we have:

(2√3)^2 = (2BC)^2 + BC^2 - 2(2BC) * BC * cos(A)

Simplifying the equation:

12 = 4BC^2 + BC^2 - 4BC^2 * cos(A)

12 = BC^2 - 4BC^2 * cos(A)

Rearranging the equation:

4BC^2 * cos(A) = BC^2 - 12

Dividing both sides by BC^2:

4cos(A) = 1 - 12/BC^2

Substituting AB = 2BC:

4cos(A) = 1 - 12/(4BC^2)

4cos(A) = 1 - 3/BC^2

cos(A) = (1 - 3/BC^2)/4

cos(A) = (BC^2 - 3)/(4BC^2)

Now, we can substitute the value of BC from the given information:

cos(A) = (BC^2 - 3)/(4(AB/2)^2)

cos(A) = (BC^2 - 3)/(4(BC/2)^2)

cos(A) = (BC^2 - 3)/(BC^2)

cos(A) = 1 - 3/BC^2

cos(A) = 1 - 3/(BC^2)

Now, we can substitute the value of BC from the given information:

cos(A) = 1 - 3/(BC^2)

cos(A) = 1 - 3/(4BC^2)

cos(A) = 1 - 3/16BC^2

cos(A) = (16BC^2 - 3)/(16BC^2)

Now, we can substitute the value of BC from the given information:

cos(A) = (16(AC/2)^2 - 3)/(16(AC/2)^2)

cos(A) = (16(AC^2)/4 - 3)/(16(AC^2)/4)

cos(A) = (4AC^2 - 3)/(4AC^2)

cos(A) = 1 - 3/(4AC^2)

cos(A) = 1 - 3/(4(2√3)^2)

cos(A) = 1 - 3/(4(12))

cos(A) = 1 - 3/48

cos(A) = 1 - 1/16

cos(A) = 15/16

To find angle A, we can take the inverse cosine (arccos) of cos(A):

A = arccos(15/16)

Using a calculator, we find that A ≈ 22.62 degrees.

Answer:

The measure of angle A is approximately 22.62 degrees.