解答这题---**Extraction Content:**
**Question Stem:**
习题 3. 如图, 在△ABC 中, ∠BAC = 120°, AC = 6, AB = 4, 则 BC 的长为______
**Mathematical Information:**
Triangle ABC
∠BAC = 120°
AC = 6
AB = 4
Find the length of BC.
**Geometric Figure Description:**
* Type: Triangle
* Main Elements:
* Three vertices labeled A, B, and C.
* Three sides forming the triangle.
* Vertex A is positioned roughly in the upper middle.
* Vertex B is positioned to the left of A.
* Vertex C is positioned to the lower right relative to A and B.
* Side AB is labeled with the number "4".
* Side AC is labeled with the letter "b" and the number "6" is written next to it.
* The angle at vertex A (∠BAC) is labeled with "120°".
* Side BC is labeled with the letter "a".
* The side length 4 is written near side AB.
* The side length 6 is written near side AC.
**Options:**
No options are visible in the provided image.
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答案文本
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We have triangle ABC where angle BAC equals 120 degrees, side AC equals 6, and side AB equals 4. We need to find the length of side BC. This is a perfect application of the Law of Cosines, which relates the sides and angles of any triangle.
The Law of Cosines is the key to solving this problem. It states that c squared equals a squared plus b squared minus 2ab cosine of angle C. In our case, BC squared equals AB squared plus AC squared minus 2 times AB times AC times cosine of angle BAC. This formula works for any triangle, not just right triangles.
Now let's substitute our known values into the Law of Cosines formula. We have AB equals 4, AC equals 6, and angle BAC equals 120 degrees. First, we calculate 4 squared equals 16, and 6 squared equals 36. The cosine of 120 degrees equals negative one half. So we get BC squared equals 16 plus 36 minus 48 times negative one half, which equals 52 plus 24, giving us BC squared equals 76.
Now we need to find BC by taking the square root of 76. We can simplify this by factoring 76 as 4 times 19. The square root of 76 equals the square root of 4 times 19, which equals the square root of 4 times the square root of 19. Since the square root of 4 is 2, we get BC equals 2 times the square root of 19. Therefore, the length of side BC is 2 root 19.
Let's summarize our solution. We used the Law of Cosines to find the length of side BC in triangle ABC. Given that angle BAC equals 120 degrees, AB equals 4, and AC equals 6, we calculated BC squared equals 76, which gives us BC equals 2 root 19. This completes our solution to the triangle problem.