Differences between AB and AC in Triangle ABC with BC 4.5 cm and Angle C 60°

Introduction

Given the construction of a triangle ABC with BC 4.5 cm and angle C 60°, this article discusses the method to determine the difference between the lengths of sides AB and AC. The process involves the application of the Law of Sines and an understanding of triangle properties. This approach provides a step-by-step guide to solving the problem and includes a detailed mathematical analysis.

Step-by-Step Solution

Step 1: Applying the Law of Sines

The Law of Sines states the following equation for a triangle:

(frac{a}{sin A} frac{b}{sin B} frac{c}{sin C})

Here, (a AC), (b AB), and (c BC 4.5 text{cm}). Given that (angle C 60^circ), we can write:

(frac{a}{sin A} frac{b}{sin B} frac{4.5}{sin 60^circ})

Step 2: Calculating the Ratio of the Sides

Since (sin 60^circ frac{sqrt{3}}{2}), we have:

(frac{a}{sin A} frac{b}{sin B} frac{4.5}{frac{sqrt{3}}{2}} frac{9}{sqrt{3}} 3sqrt{3})

Step 3: Expressing AC and AB

This can be written as:

(a AC 3sqrt{3} sin A)

(b AB 3sqrt{3} sin B)

Step 4: Finding the Difference AB - AC

To find the difference:

(AB - AC 3sqrt{3} sin B - 3sqrt{3} sin A 3sqrt{3} (sin B - sin A))

Step 5: Angle Relationships

Using the triangle angle sum property:

(A B C 180^circ Rightarrow A B 120^circ)

Thus, we can express B in terms of A:

(B 120^circ - A)

Step 6: Substituting B

Substitute B:

(AB - AC 3sqrt{3} (sin 120^circ - A - sin A))

Using the sine subtraction formula:

(sin 120^circ - A sin 120^circ cos A - cos 120^circ sin A frac{sqrt{3}}{2} cos A - frac{1}{2} sin A)

Step 7: Calculating the Difference

Substitute the sine subtraction formula:

(AB - AC 3sqrt{3} left(frac{sqrt{3}}{2} cos A - frac{1}{2} sin Aright))

This simplifies to: