Question

If two sides of a triangle have the following measures, find the range of possible measures for the third side. 12,8

Short answer

The third side must be greater than 4 and less than 20.

Step-by-step solution

Understanding Triangle Inequality

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This theorem helps us find possible lengths for the third side when two sides are known.

Set Up the Inequalities

Let's denote the length of the third side by \( x \). Use the Triangle Inequality Theorem to set up the following inequalities:1. \( 12 + 8 > x \)2. \( 12 + x > 8 \)3. \( 8 + x > 12 \)

Solve Each Inequality

Solve each inequality to find the range of possible values for \( x \):1. \( 12 + 8 > x \) simplifies to \( x < 20 \).2. \( 12 + x > 8 \) simplifies to \( x > -4 \). Since a side length cannot be negative, this provides no restriction.3. \( 8 + x > 12 \) simplifies to \( x > 4 \).

Combine the Results

Now we combine the results we have from solving the inequalities:- The valid range for \( x \) given that it must satisfy \( x < 20 \) and \( x > 4 \) is \( 4 < x < 20 \). This means the third side of the triangle has to be greater than 4 and less than 20.

Key concepts

Understanding Geometry and Triangles

Geometry is the branch of mathematics that studies the sizes, shapes, properties, and dimensions of objects and spaces. It's like a mathematical playground where we explore the different ways things can fit together. One of the simplest shapes in geometry is the triangle. The triangle is a three-sided polygon that forms the basis for many geometric concepts. Triangles can vary widely—some are right-angled, and others are equilateral, scalene, or isosceles. Each type of triangle has its own unique properties and internal angles.
Triangles are special because they always have three sides and three angles, with the sum of the angles always equaling 180 degrees. This is a basic yet important principle in geometry. Understanding this can help us solve many problems involving triangles!

Exploring Triangle Sides

The sides of a triangle play a crucial role in determining the shape and type of triangle. The sides are often labeled with letters like 'a,' 'b,' and 'c.' The lengths of these sides determine the triangle's properties, for instance:

• If all three sides are equal, it's called an equilateral triangle.
• If two sides are equal, it's known as an isosceles triangle.
• If all three sides are different, we call it a scalene triangle.

In the given problem, we know two sides of a triangle: 12 and 8. We need to figure out the possible range for the third side. The nature of the triangle can change vastly depending on this third side. This leads us into the realm of inequalities, where we find the suitable range for the unknown side.

Understanding Inequalities in Triangles

In mathematics, inequalities express the relationship between different values. They tell us how one number is larger or smaller than another. The Triangle Inequality Theorem is a special inequality used in geometry.
This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Let's see how it applies to the problem at hand. We have two sides: 12 and 8, and we need to find the range for the third side, which we'll call 'x.' According to the Triangle Inequality Theorem:

• The sum of the two known sides (12 + 8) must be greater than 'x,' so: \( 12 + 8 > x \).
• One known side plus the unknown side must be greater than the other known side, which gives us two more inequalities:
  • \( 12 + x > 8 \)
  • \( 8 + x > 12 \)

If you solve these inequalities, you find that the third side should be greater than 4 and less than 20. This means that the third side must satisfy \( 4 < x < 20 \). This range ensures that a valid triangle can be formed.