Math Problem Statement

已知△ABC的内角A,B,C的对边分别为a,b,c,且a sin(A + C)/2 = b sin A。 (1) 求 B; (2) 若△ABC为锐角三角形,且 h_b = √3,求△ABC周长的取值范围。

Solution

To solve this problem, we have a triangle ABC\triangle ABC with internal angles AA, BB, and CC, and opposite sides aa, bb, and cc respectively. We are given two tasks:

  1. Find BB.
  2. Given that ABC\triangle ABC is an acute triangle and hb=3h_b = \sqrt{3} (where hbh_b is the altitude from vertex BB to side bb), determine the range of values for the perimeter of ABC\triangle ABC.

Let's go through each part systematically.

Part 1: Find BB

From the information given, we know: asinA+C2=bsinAa \sin \frac{A + C}{2} = b \sin A This equation can be used to find a relationship between the angles or the sides of the triangle.

Rewrite this equation by using trigonometric identities, particularly the half-angle formulas. This would allow us to express BB in terms of other known angles and potentially find a unique solution for BB.

Part 2: Find the Range of the Perimeter of ABC\triangle ABC when hb=3h_b = \sqrt{3}

Since hb=3h_b = \sqrt{3}, we can express hbh_b in terms of side bb and angle AA as: hb=bsinA=3h_b = b \sin A = \sqrt{3} From here, you could solve for bb in terms of AA. Then, using the conditions of an acute triangle, you would analyze the feasible values of aa, bb, and cc to calculate the perimeter and determine its range.

Would you like me to go through the detailed algebraic steps for each part?


Here are some relative questions to consider further:

  1. How can we apply the Law of Sines in triangle ABCABC to find the other angles?
  2. What trigonometric identities are useful when working with acute triangles?
  3. How does the condition hb=3h_b = \sqrt{3} limit the values of aa, bb, and cc?
  4. Could the triangle inequality provide additional constraints for the sides?
  5. How would the result change if ABC\triangle ABC was obtuse instead of acute?

Tip

Remember, understanding the properties of special triangles, such as those with specific altitude relationships, can simplify calculations significantly.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Triangle Properties
Acute Triangles

Formulas

a sin((A + C)/2) = b sin A
h_b = b sin A

Theorems

Sine Rule
Triangle Inequality

Suitable Grade Level

Grades 10-12