Solving for the Sides of a Triangle with Given Conditions

Understanding and solving for the sides of a triangle based on given conditions is an essential part of geometry and trigonometry. In this article, we will explore how to determine the lengths of the sides of a triangle when the perimeter and the relationship between the sides are known. Specifically, we will solve for a triangle where one side is given, another side is twice the third side, and the perimeter is 18 cm.

Problem Statement

We are given a triangle with the following properties:

The first side (a) is 6 cm. The second side (b) is twice the length of the third side (c). The perimeter of the triangle is 18 cm.

Step-by-Step Solution

Define the variables:

a 6 cm (First side)

b 2c (Second side, twice the third side)

c Third side

Formulate the perimeter equation:

The perimeter is the sum of all sides:

a b c 18 Substitute the given and derived values:

Substitute the value of a and the relationship between b and c into the perimeter equation:

6 2c c 18 Combine like terms and solve for c:

Combining the terms with c:

6 3c 18

Subtract 6 from both sides:

3c 12

Divide both sides by 3:

c 4 Find the value of b:

Now that c is known, we can find b:

b 2c 2 × 4 8 Summarize the side lengths:

The sides of the triangle are:

First side (a) 6 cm Second side (b) 8 cm Third side (c) 4 cm

Explanation and Recap

The sides of the triangle can be determined by setting up an algebraic equation based on the given conditions and solving it step by step. The problem required us to use the properties of the triangle (perimeter and side relationships) to form an equation and solve it to find the unknown side lengths.

Useful Algebraic Equation

A useful algebraic equation that we can derive from the problem is:

a 2c c 18

where:

a is the first side (given as 6 cm) c is the third side (unknown) 2c is the second side (twice the third side)

Conclusion

By solving the algebraic equation, we were able to determine that the sides of the triangle are 6 cm, 8 cm, and 4 cm. This problem demonstrates the importance of understanding basic algebra in solving geometric problems.

Further Reading

For more information on similar problems and geometric concepts, refer to:

Math Warehouse Britannica: Triangle in Geometry