Determine if Triangle ABC with Sides 2X and 12-X is Equilateral

Q: How can I tell the difference between equilateral, isosceles, and scalene triangles?

Equilateral: All 3 sides equal. Isosceles: Exactly 2 sides equal. Scalene: No sides equal. Always calculate the actual side lengths to classify correctly!

Q: Why do I need to set 2X equal to 12-X instead of just looking at the diagram?

Diagrams aren't drawn to scale! The sides might look equal, but you need algebra to prove it. Only by solving $ 2X = 12 - X $ can you find the exact value that makes them equal.

Equilateral Triangles with Variable Expressions

Given the size of the 3 sides of the triangle, is it an equilateral triangle?

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Step-by-step video solution

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00:00 Determine whether the triangle is equilateral

00:03 Let's treat it as equilateral, determine the value of X and then finally verify

00:07 Proceed to equate the two sides and determine X

00:11 Isolate X

00:16 This is the value of X assuming the sides are equal, let's not proceed to verify this

00:19 Substitute X into each side and check whether all the sides are equal

00:31 This is the value of one side

00:34 Substitute into the second side

00:38 The same once again for the third side, the triangle is indeed equilateral

00:41 This is the solution

Step-by-step written solution

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Understand the problem

Given the size of the 3 sides of the triangle, is it an equilateral triangle?

Step-by-step solution

To determine if the triangle is equilateral, we need to check if all three sides of the triangle are equal.

The given side lengths are $2X$ , $12 - X$ , and $12 - X$ .

For the triangle to be equilateral, we must have the equality:

$2X = 12 - X$

Let's solve this equation:

$\begin{aligned} 2X &= 12 - X \\ 2X + X &= 12 \\ 3X &= 12 \\ X &= \frac{12}{3} \\ X &= 4 \end{aligned}$

Substitute $X = 4$ back into the expressions for the sides:

$2X = 2(4) = 8$
$12 - X = 12 - 4 = 8$
The third side, also $12 - X = 8$ .

The three calculated side lengths are $8$ , $8$ , and $8$ .

Since all three sides are equal, the triangle is an equilateral triangle.

Therefore, the answer is Yes, the triangle is equilateral.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Definition: All three sides must be equal for equilateral triangles
Method: Set all sides equal: $2X = 12 - X$
Verify: Substitute X = 4 back: 2(4) = 8 and 12 - 4 = 8 ✓

Common Mistakes

Avoid these frequent errors

Assuming the triangle is equilateral without checking
Don't just look at the diagram and guess! The triangle could be isosceles (only two equal sides) or scalene. Always solve algebraically by setting all sides equal to find X, then verify all three sides are actually equal.

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

How can I tell the difference between equilateral, isosceles, and scalene triangles?

Equilateral: All 3 sides equal. Isosceles: Exactly 2 sides equal. Scalene: No sides equal. Always calculate the actual side lengths to classify correctly!

What if I get a negative value for X when I solve?

Check if the triangle can actually exist! Side lengths must be positive. If X gives negative sides, or if the triangle inequality fails, then no triangle exists with those expressions.

Why do I need to set 2X equal to 12-X instead of just looking at the diagram?

Diagrams aren't drawn to scale! The sides might look equal, but you need algebra to prove it. Only by solving $2X = 12 - X$ can you find the exact value that makes them equal.

Do I need to check if the third side is also equal?

Yes! Even though two sides have the same expression (12-X), you must verify that when X = 4, all three sides equal the same value: 8, 8, and 8.

What if the equation gives me a decimal or fraction for X?

That's fine! Just substitute it back carefully. For example, if X = 2.5, then $2X = 5$ and $12 - X = 9.5$ . Since 5 ≠ 9.5, it wouldn't be equilateral.

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