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Have you ever been given three lengths and wondered if you could actually form a triangle with them? It turns out, there's a simple mathematical rule that gives you the answer every time! In this video, we break down one of the most fundamental concepts in geometry: The Triangle Inequality Theorem.

We'll guide you step-by-step through the logic, using the exact problems and visual explanations from your textbook. By the end of this tutorial, you'll be able to quickly determine if any three side lengths can form a triangle.
What is the Triangle Inequality Theorem? 🤔
For any triangle to exist, the sum of the lengths of any two sides must be greater than the length of the third side.

If you have a triangle with side lengths a, b, and c, then all three of the following conditions must be true:
a+b greater than c

a+c greater than b

b+c greater than a

The Shortcut: You only need to check one thing! Add the two shorter lengths together. If their sum is greater than the longest length, a triangle can be formed.
The Shortcut: You only need to check one thing! Add the two shorter lengths together. If their sum is greater than the longest length, a triangle can be formed.
Let's Solve an Example from the Video
Problem: Can the lengths 5, 5, 8 form a triangle?

Identify the lengths: a=5, b=5, c=8.

Find the two shorter sides: 5 and 5.

Add them together: 5+5=10.

Compare to the longest side 8

Answer: Yes! A triangle can be formed.

Problem: Can the lengths 10, 20, 35 form a triangle?

Identify the lengths: a=10, b=20, c=35.

Find the two shorter sides: 10 and 20.

Add them together: 10+20=30.

Compare to the longest side 35

Answer: No! A triangle cannot be formed with these lengths.

The Circle Analogy Explained ✍️
Imagine the longest side is a fixed line segment. The other two sides can be thought of as the radii of two circles, with their centers at the ends of that line segment.

If the sum of the radii is greater than the segment length, the circles will intersect at two points, creating the third vertex of the triangle.

If the sum of the radii is equal to the segment length, the circles will touch at one point, forming a flat line (a degenerate triangle).

If the sum of the radii is less than the segment length, the circles will not intersect at all, and it's impossible to form a triangle.

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