Equilateral Triangle Verification: Comparing Sides 10+X, 5X-10, and 3X

Q: How do I know which two expressions to set equal?

You need all three sides to be equal! Since two sides are already the same (10 + X), set one of them equal to the third side: $ 10 + X = 5X - 10 $.

Q: What if X gives me negative side lengths?

Great question! Side lengths must be positive. After finding X = 5, check: 10 + 5 = 15 ✓ and 5(5) - 10 = 15 ✓. All positive, so it's valid!

Q: Why can't I just assume it's equilateral from the diagram?

Never trust diagrams alone! They can be misleading. Always solve algebraically to prove whether the triangle is truly equilateral.

Q: What does it mean if I get different values for X?

If setting different pairs of sides equal gives different X values, then the triangle cannot be equilateral for any value of X. The sides will never all be equal.

Equilateral Triangle Properties with Algebraic Side Lengths

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:13 Let's check if the triangle is equilateral.

00:17 Assume it's equilateral, find X, then verify.

00:22 Set the two sides equal and find X.

00:26 Isolate the variable X.

00:28 This is X, assuming sides are equal. Let's verify!

00:33 Plug in X to each side and check if all sides match.

00:44 Here's the length of one side.

00:49 Substitute in the second side; same result!

00:53 The third side matches too. It's equilateral!

00:58 And that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

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 given triangle is equilateral, we need to ensure that all sides are equal. Given the sides:

$a = 10 + X$
$b = 10 + X$
$c = 5X - 10$

Since two sides are already equal ( $a = b$ ), we need to check if this is equal to the third side:

Set the side lengths $a$ and $c$ equal:

$10 + X = 5X - 10$

Solve for $X$ :

$10 + X = 5X - 10$

Subtract $X$ from both sides:

$10 = 4X - 10$

Add 10 to both sides:

$20 = 4X$

Divide by 4:

$X = 5$

Substitute $X = 5$ back into the side length expressions:

$10 + X = 10 + 5 = 15$
$5X - 10 = 5(5) - 10 = 25 - 10 = 15$

All sides are equal to 15. Hence, the triangle is equilateral.

Therefore, the solution to the problem is Yes.

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 equal expressions: 10 + X = 5X - 10 to find X = 5
Verification: Substitute X = 5 into all sides: 15 = 15 = 15 ✓

Common Mistakes

Avoid these frequent errors

Only comparing two sides instead of all three
Don't just check if two sides are equal and assume equilateral = wrong conclusion! Two equal sides make isosceles, not equilateral. Always verify all three sides equal the same value by solving for X first.

Practice Quiz

Test your knowledge with interactive questions

Is the triangle in the drawing a right triangle?

FAQ

Everything you need to know about this question

How do I know which two expressions to set equal?

You need all three sides to be equal! Since two sides are already the same (10 + X), set one of them equal to the third side: $10 + X = 5X - 10$ .

What if X gives me negative side lengths?

Great question! Side lengths must be positive. After finding X = 5, check: 10 + 5 = 15 ✓ and 5(5) - 10 = 15 ✓. All positive, so it's valid!

Why can't I just assume it's equilateral from the diagram?

Never trust diagrams alone! They can be misleading. Always solve algebraically to prove whether the triangle is truly equilateral.

What does it mean if I get different values for X?

If setting different pairs of sides equal gives different X values, then the triangle cannot be equilateral for any value of X. The sides will never all be equal.

Do I need to check the triangle inequality too?

For this problem, no! Since we found X = 5 makes all sides equal to 15, and 15 > 0, the triangle automatically satisfies the triangle inequality.

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