How to Find the Measure of Side AB in Triangle ABC

Understanding the cosine rule is essential when working with triangles and calculating the length of a side. In a triangle ABC, given the lengths of sides BC and CA, and the measure of angle C, we can determine the length of the remaining side AB. This article will guide you through the steps and illustrate the application of the cosine rule.

Introduction to the Cosine Rule

The cosine rule, also known as the law of cosines, is a formula used to find the length of a side in a triangle given the lengths of the other two sides and the included angle. The formula is given by:

a2 b2 c2 - 2ab cos(C)

Here, a, b, and c represent the lengths of the sides opposite angles A, B, and C, respectively. The angle C is the angle included between sides b and c.

Step-by-Step Guide to Using the Cosine Rule

Identify the Given Information:
Contact your given measurements. For triangle ABC with sides BC (let's call it c) and CA (let's call it b) and the angle C. The unknown side is AB (let's call it a). Apply the Cosine Rule:
Substitute the known values into the formula a2 b2 c2 - 2bc cos(C). Solve for the Unknown Side:
Once you have the equation, solve for a2. Then, take the square root to find the length of AB.

Example Application

Suppose in triangle ABC, we have:

BC 6 units (side c) CA 8 units (side b) Angle C 60°

Solution Using the Cosine Rule

Using the cosine rule:

a2 82 62 - 2(8)(6) cos(60°)

a2 64 36 - 96(0.5)

a2 100 - 48

a2 52

a √52 ≈ 7.21 units

PRACTICE PROBLEMS

In triangle DEF, with DE 5 units, EF 7 units, and angle F 120°, find the length of DF. In triangle GHI, with GH 9 units, HI 5 units, and angle H 30°, find the length of GI. In triangle JKL, with JK 10 units, KL 8 units, and angle L 90°, find the length of JL.

Conclusion

The cosine rule is a powerful tool in trigonometry, useful for finding the length of a side in a triangle when you have the lengths of the other two sides and the included angle. By mastering the application of the cosine rule, you can solve complex problems in geometry with ease. Practice with various examples to deepen your understanding and proficiency.