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

Test your knowledge with interactive questions

Choose the appropriate triangle according to the following:

Angle B equals 90 degrees.

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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