Introduction to the Internal Bisector in a Triangle

In this article, we delve into the method of calculating the length of the internal bisector in a specific triangle, utilizing various geometric and trigonometric principles. Understanding the internal bisector in a triangle not only enhances our geometric knowledge but also paves the way for solving complex problems in geometry and trigonometry.

Problem Statement

Consider a triangle (ABC) where the internal bisector of the angle (A) meets (BC) at point (D). Given the sides (AB 4), (AC 3), and angle (angle A 60°), we aim to find the length of the internal bisector (AD).

Understanding the Angle Bisector and Area Calculations

The internal bisector (AD) divides the base (BC) in the ratio given by the Angle Bisector Theorem. This theorem states that the internal bisector of an angle in a triangle divides the opposite side into segments that are proportional to the adjacent sides. Therefore, (frac{BD}{DC} frac{AB}{AC} frac{4}{3}). We denote the length of (BC) as (a), such that (a BD DC) and (BD frac{4a}{7}) and (DC frac{3a}{7}).

Trigonometric Approach to Area and Length of AD

The area of triangle (ABC) can be determined using the trigonometric formula:

[ [ABC] frac{1}{2} cdot AB cdot AC cdot sin(angle A) frac{1}{2} cdot 4 cdot 3 cdot frac{sqrt{3}}{2} 3sqrt{3} ]

Since (AD) is the internal bisector of angle (A), it divides triangle (ABC) into two sub-triangles (ABD) and (ACD). The area of these sub-triangles can be calculated as:

[ [ABD] frac{1}{7} [ABC] frac{3sqrt{3}}{7} ]

Using trigonometry, we find the length of (AD) as follows:

[ frac{1}{2} cdot AB cdot AD cdot frac{1}{2} frac{3sqrt{3}}{7} ]

From which:

[ AD frac{12sqrt{3}}{7} approx 2.97 ]

Using Trigonometric and Cevian Formulas

Alternatively, to find the length of (AD), we apply the sine rule and the triangle area formula. Let’s explore two straightforward methods:

Method 1: Using the Sine Rule and Cevian Formulas

Applying the sine rule to triangle (ADB) and (ABC) gives:

[ frac{BD}{sin DAB} frac{AB}{sin ADB} frac{AD}{sin ABD} ]

We also have:

[ frac{AC}{sin ABC} frac{BC}{sin 60°} frac{3}{frac{sqrt{3}}{2}} ]

Therefore, we can express (AD) in terms of the sides and angles as:

[ AD frac{3 sqrt{3}}{2} cdot frac{AD}{2BD} ]

By solving, we obtain:

[ AD frac{6sqrt{3}}{7} approx 2.96923 ]

Method 2: Using the Standard Angle Bisector Length Formula

Another method involves using the standard formula for the length of the angle bisector:

[ AD frac{2bc cos frac{A}{2}}{b c} ]

Substituting (b 4, c 3, A 60°):

[ AD frac{2 cdot 4 cdot 3 cdot cos 30°}{4 3} frac{12sqrt{3}}{7} approx 2.96923 ]

Verification Using Geometric and Trigonometric Techniques

To verify the results, we can use the triangle area formula and the lengths of the sides. The area of triangle (ABC) can be calculated as:

[ [ABC] frac{1}{2} cdot AB cdot AC cdot sin 60° frac{1}{2} cdot 4 cdot 3 cdot frac{sqrt{3}}{2} 3sqrt{3} ]

The area of sub-triangles (ABD) and (ACD) can be written in terms of (AD):

[ [ABD] [ACD] 2 cdot frac{1}{2} cdot AB cdot AD cdot frac{sin 60°}{2} 3sqrt{3} cdot frac{2}{7} frac{12sqrt{3}}{7} ]

This confirms the length of (AD) as:

[ AD frac{12sqrt{3}}{7} approx 2.96 ]

Generalized Formula and Conclusion

In a more general scenario, if the angle bisector divides angle (A) into (alpha) and (beta), the length of the internal bisector (AD) can be given by:

[ AD frac{AB cdot AC cdot sin A}{AB cdot sin alpha cdot AC cdot sin beta} ]

For the given problem, setting (AB 4, AC 3, A 60°, alpha beta 30°), we find:

[ AD frac{4 cdot 3 cdot frac{sqrt{3}}{2}}{4 cdot frac{1}{2} cdot 3 cdot frac{1}{2}} frac{12sqrt{3}}{7} approx 2.96229 ]

In summary, understanding and applying the internal bisector formula, area calculations, and trigonometric principles allow us to accurately find the length of the internal bisector in a given triangle, showcasing the elegance and interconnectedness of geometric and trigonometric concepts.