Triangle Side Analysis: Determining if a, 2a-a, and 3a Form a Scalene Triangle

Q: Why do I need to simplify 2a-a first?

Algebraic expressions can look different but have the same value! $ 2a - a = a $, so you're really comparing a, a, and 3a - not three different expressions.

Q: What makes a triangle scalene vs isosceles?

A scalene triangle has all three sides different lengths. An isosceles triangle has exactly two equal sides. Since we have sides $ a, a, 3a $, this is isosceles!

Q: How do I remember the triangle types?

Scalene: All sides different (like a scale with unequal weights)Isosceles: Two sides equal (sounds like "I saw two legs")Equilateral: All sides equal

Q: Does this triangle satisfy the triangle inequality?

Yes! For any positive value of a: $ a + a = 2a is false, but \( a + 3a = 4a > a $ and $ a + 3a = 4a > a $ are both true. Wait - we need $ a + a > 3a $, so $ 2a > 3a $, which means $ a $ must be negative! This only works for negative values of a.

Triangle Classification with Algebraic Side Lengths

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

Watch the teacher solve the problem with clear explanations

00:00 Determine whether the triangle is scalene

00:02 Calculate the side length according to the given data

00:05 Side lengths according to the given data

00:08 The triangle is not scalene, and this is the solution to the question

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 determine the type of triangle based on the given side lengths, we proceed as follows:

Simplify the expressions for the side lengths:

The first side is $a$ .
The second side simplifies as $2a-a = a$ .
The third side is $3a$ .

Compare the lengths to determine equality:

The first side is $a$ , and the second side is also $a$ .
The third side is $3a$ , which is different from the first two sides ( $a$ ).

Since the sides $a$ , $a$ , and $3a$ have two sides that are equal, the triangle is not a scalene triangle, which has all sides of different lengths.

Therefore, the triangle is not a triangle with different sides (scalene triangle). The correct answer is "No".

Final Answer

Key Points to Remember

Essential concepts to master this topic

Simplification: Always simplify algebraic expressions first: $2a - a = a$
Classification: Compare simplified sides: $a, a, 3a$ shows two equal sides
Verification: Check triangle type: two equal sides means isosceles, not scalene ✓

Common Mistakes

Avoid these frequent errors

Not simplifying algebraic expressions before comparing
Don't compare a, 2a-a, and 3a directly without simplifying = wrong triangle type! The expression 2a-a looks different from a, but they're equal. Always simplify all algebraic expressions first, then compare the actual values.

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 simplify 2a-a first?

Algebraic expressions can look different but have the same value! $2a - a = a$ , so you're really comparing a, a, and 3a - not three different expressions.

What makes a triangle scalene vs isosceles?

A scalene triangle has all three sides different lengths. An isosceles triangle has exactly two equal sides. Since we have sides $a, a, 3a$ , this is isosceles!

Could this triangle have different side lengths if a changes?

No! No matter what value a takes, you'll always have two sides of length $a$ and one side of length $3a$ . The ratio stays the same.

How do I remember the triangle types?

Scalene: All sides different (like a scale with unequal weights)
Isosceles: Two sides equal (sounds like "I saw two legs")
Equilateral: All sides equal

Does this triangle satisfy the triangle inequality?

Yes! For any positive value of a: $a + a = 2a < 3a$ is false, but $a + 3a = 4a > a$ and $a + 3a = 4a > a$ are both true. Wait - we need $a + a > 3a$ , so $2a > 3a$ , which means $a$ must be negative! This only works for negative values of a.

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