๐ Topic Summary
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. In simpler terms, if you have sides of lengths $a$, $b$, and $c$, then the following inequalities must all be true: $a + b > c$, $a + c > b$, and $b + c > a$. If any of these inequalities is not true, then you cannot form a triangle with those side lengths. This theorem helps determine if a triangle can even exist given three side lengths.
Let's illustrate with an example: Suppose we have sides of lengths 3, 4, and 5. Then, $3 + 4 > 5$, $3 + 5 > 4$, and $4 + 5 > 3$. Since all three inequalities hold, a triangle can be formed with these side lengths. Now, consider sides of lengths 1, 2, and 5. Here, $1 + 2$ is not greater than $5$, so a triangle cannot be formed.
๐ค Part A: Vocabulary
- ๐ Term: Triangle Inequality Theorem
๐ก Definition: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
- ๐ Term: Side Length
๐ Definition: The distance from one vertex of a triangle to another.
- โ Term: Sum
๐ข Definition: The result of adding two or more numbers together.
- โณ Term: Triangle
๐ Definition: A closed figure formed by three line segments.
- โ Term: Inequality
โ๏ธ Definition: A mathematical statement that compares two expressions using symbols like >, <, โฅ, or โค.
โ๏ธ Part B: Fill in the Blanks
The Triangle Inequality Theorem states that the _______ of any two sides of a triangle must be _______ than the third side. If the sum of two sides is _______ than the third side, then a triangle _______ be formed. To check if three given side lengths can form a triangle, you must verify that all three possible _______ hold true.
Word Bank: inequalities, sum, greater, less, cannot
๐ค Part C: Critical Thinking
Can you explain in your own words why the Triangle Inequality Theorem is true? Provide an example to support your explanation.