Triangle Side Length Analysis: Comparing a, a-2, and a+1

Q: Why do I need to check three different inequalities?

The triangle inequality theorem requires that every pair of sides must sum to more than the third side. If even one fails, no triangle can exist with those measurements!

Q: What does it mean for a triangle to have 'different sides'?

A triangle with different sides is called scalene. This means no two sides are equal in length - each side has a unique measurement.

Q: How do I know what values of 'a' make this work?

From the triangle inequalities, you need $a > 3$. This ensures the triangle exists and all sides are different since $a ≠ a-2 ≠ a+1$.

Q: Can I just assume the triangle exists if it's drawn?

No! Always verify mathematically. A diagram might show a triangle, but the algebraic expressions for the sides must satisfy the triangle inequality for the triangle to actually exist.

Q: What happens if a = 3 exactly?

When $a = 3$, the sides become 3, 1, and 4. Check: $3 + 1 = 4$, which violates the triangle inequality (sum equals, doesn't exceed). No triangle can form!

Triangle Inequality with Variable Expressions

Given the values of the sides of a triangle, is it a triangle with different sides?

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Step-by-step video solution

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00:00 Determine whether the triangle is scalene

00:02 The side lengths according to the given data

00:05 All the side lengths are different, therefore the triangle is scalene

00:08 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the values of the sides of a triangle, is it a triangle with different sides?

Step-by-step solution

To solve this problem, we'll determine whether a triangle with side lengths $a$ , $a-2$ , and $a+1$ is scalene:

Step 1: Verify the triangle inequality theorem.
- Check $a + (a-2) > (a+1)$ : $2a - 2 > a + 1$ simplifies to $a > 3$ . - Check $(a-2) + (a+1) > a$ : $(2a - 1) > a$ simplifies to $a > 1$ . - Check $a + (a+1) > (a-2)$ : $2a + 1 > a - 2$ simplifies to $a > -3/2$ , which is always true for $a > 2$ .
Step 2: Check if all sides are different.
- Compare $a \neq a-2$ : True, always holds as $a > 2$ .
- Compare $a \neq a+1$ : True, always holds.
- Compare $a-2 \neq a+1$ : True, simplifies to $a \neq 3$ , which holds since $a > 3$ .

All side lengths satisfy the triangle inequality and are different. Therefore, the triangle is scalene. The solution to the problem is "Yes," this is a triangle with different sides.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Triangle Inequality: Sum of any two sides must exceed the third
Technique: Check $a + (a-2) > (a+1)$ gives $a > 3$
Verification: Compare all three sides to confirm they're different when $a > 3$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to check if triangle inequality holds for all three combinations
Don't just check one inequality like $a > (a-2)$ = obvious but incomplete! This misses critical constraints. Always verify all three triangle inequalities: each pair of sides must sum to more than the third side.

Practice Quiz

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In a right triangle, the side opposite the right angle is called....?

FAQ

Everything you need to know about this question

Why do I need to check three different inequalities?

The triangle inequality theorem requires that every pair of sides must sum to more than the third side. If even one fails, no triangle can exist with those measurements!

What does it mean for a triangle to have 'different sides'?

A triangle with different sides is called scalene. This means no two sides are equal in length - each side has a unique measurement.

How do I know what values of 'a' make this work?

From the triangle inequalities, you need $a > 3$ . This ensures the triangle exists and all sides are different since $a ≠ a-2 ≠ a+1$ .

Can I just assume the triangle exists if it's drawn?

No! Always verify mathematically. A diagram might show a triangle, but the algebraic expressions for the sides must satisfy the triangle inequality for the triangle to actually exist.

What happens if a = 3 exactly?

When $a = 3$ , the sides become 3, 1, and 4. Check: $3 + 1 = 4$ , which violates the triangle inequality (sum equals, doesn't exceed). No triangle can form!

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