Calculating the Length of Angle Bisector CD in Triangle ABC

Given a triangle ABC with sides BCa, ACb, and the angle ACB120°, the goal is to find the length of the angle bisector CD. This problem involves understanding triangle properties and applying trigonometric principles.

Understanding the Triangle Configuration

In triangle ABC, the sides and angles are as follows:

BC a AC b angle ACB 120°

Step 1: Finding AB Using the Law of Cosines

First, we use the Law of Cosines to find the length of side AB.

The Law of Cosines states:

c^2 a^2 b^2 - 2ab cos(120°)

Since cos(120°) -1/2, we can substitute this into the equation:

c^2 a^2 b^2 ab

Therefore:

AB c √(a^2 b^2 ab)

Step 2: Using the Angle Bisector Length Formula

The length of the angle bisector CD can also be found using a specific formula:

CD (2ab)/(a b) * cos(60°/2)

Given that angle ACB is 120°, the angle bisector splits this angle into two 60° angles. Therefore:

cos(60°/2) cos(30°) √3/2

Substituting these values into the formula, we get:

CD (2ab)/(a b) * (√3/2) (ab * √3)/(a b)

Working Through the Example

Given this formula, let's break down a specific example:

Consider triangle ABC with BC a 10 units, AC b 15 units, and angle ACB 120°. Using the Law of Cosines to find AB:

AB √(10^2 15^2 10*15)

AB √(100 225 150) √475 21.8 units (approximately)

Now, calculate the length of CD:

CD (10*15 * √3)/(10 15) (150 * √3)/25 6√3 units (approximately 10.39 units)

Alternative Approach Using Triangle Area

Another way to approach this problem is by considering the areas of triangles ACD and BCD:

Area of triangle ABC (a * b * sin(120°))/2 (a * b * √3/2)/2 (a * b * √3)/4

Area of triangle ACD Area of triangle BCD (b * CD * sin(60°))/2 (a * CD * sin(60°))/2 (b * CD * √3/4)/2 (a * CD * √3/4)/2 (a * b * √3)/4

Equating the areas, we get:

b * CD * √3/4 a * CD * √3/4 (a * b * √3)/4

Simplifying:

CD * (a b) * √3/4 (a * b * √3)/4

CD (ab)/(a b)

This confirms the earlier formula for CD, validating our solution.

Conclusion: The length of the angle bisector CD in triangle ABC, where BC a, AC b, and angle ACB 120°, can be calculated using the following formula:

CD (ab)/(a b)

Additional Insights

It’s worth noting that while CD divides the triangle into two smaller triangles, the ratio of the areas of these triangles is not necessarily proportional to the sides a and b. Instead, they are determined by the angle bisector theorem and the specific geometry of the 30-60-90 triangles mentioned in the guess.

In summary, by applying both the Law of Cosines and the angle bisector length formula, we can accurately determine the length of CD, providing a robust solution to the problem at hand.