Triangle Properties Analysis: Can a 90° Right Triangle Be Equilateral?

Q: Why can't a triangle have both equal sides and a right angle?

In triangles, equal sides create equal angles. If all three sides are equal, all three angles must be $60^\circ$. But a right triangle needs one $90^\circ$ angle, which breaks this pattern!

Q: What about isosceles right triangles - aren't they close?

Isosceles right triangles have two equal sides and angles of $45^\circ-45^\circ-90^\circ$. They're right triangles but not equilateral because only two sides are equal, not all three.

Q: How do I remember the angle rules for different triangles?

Right triangle: One angle is $90^\circ$Equilateral triangle: All angles are $60^\circ$Key rule: All triangle angles must sum to $180^\circ$

Q: Can any triangle be both types of special triangles?

No triangle can be both types because their defining properties conflict. Each triangle type has exclusive requirements that cannot overlap with other special triangle types.

Q: What if I'm not sure about angle measurements?

Always check the angle sum rule: all triangle angles add to $180^\circ$. If you know one angle is $90^\circ$, the other two must sum to $90^\circ$, so they cannot both be $60^\circ$!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine whether a right triangle can be equilateral

00:04 An equilateral triangle is also equiangular

00:13 All of its angles are equal to 60

00:19 A right triangle must have an angle equal to 90 degrees

00:27 Therefore, a right triangle cannot be equilateral

00:32 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Can a right triangle be equilateral?

2

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the properties of a right triangle.
Step 2: Identify the properties of an equilateral triangle.
Step 3: Compare these properties to determine if a right triangle can be equilateral.

Now, let's work through each step:

Step 1: A right triangle is defined by having one angle equal to $90^\circ$ .
Step 2: An equilateral triangle is defined by having all three sides of equal length and all three angles equal to $60^\circ$ .
Step 3: Compare the angle measurements: A right triangle cannot have all angles $60^\circ$ because it requires one angle to be $90^\circ$ . Likewise, an equilateral triangle cannot have a $90^\circ$ angle, as all its angles must be $60^\circ$ .

Therefore, it is impossible for a right triangle to be equilateral, as they fundamentally differ in angle requirements.

The answer to the problem is No.

3

Final Answer

No

Key Points to Remember

Essential concepts to master this topic

Definition: Right triangles have one 90° angle requirement
Comparison: Equilateral triangles have three 60° angles exactly
Check: Compare angle sums: 90° + 2angles vs 60° + 60° + 60° = 180° ✓

Common Mistakes

Avoid these frequent errors

Confusing side lengths with angle measures
Don't think equal sides automatically mean equal angles in all triangles! This ignores the fundamental angle constraint of right triangles. Always check angle requirements first: right triangles need exactly one $90^\circ$ angle, while equilateral triangles need three $60^\circ$ angles.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why can't a triangle have both equal sides and a right angle?

+

In triangles, equal sides create equal angles. If all three sides are equal, all three angles must be $60^\circ$ . But a right triangle needs one $90^\circ$ angle, which breaks this pattern!

What about isosceles right triangles - aren't they close?

+

Isosceles right triangles have two equal sides and angles of $45^\circ-45^\circ-90^\circ$ . They're right triangles but not equilateral because only two sides are equal, not all three.

How do I remember the angle rules for different triangles?

+

Right triangle: One angle is $90^\circ$
Equilateral triangle: All angles are $60^\circ$
Key rule: All triangle angles must sum to $180^\circ$

Can any triangle be both types of special triangles?

+

No triangle can be both types because their defining properties conflict. Each triangle type has exclusive requirements that cannot overlap with other special triangle types.

What if I'm not sure about angle measurements?

+

Always check the angle sum rule: all triangle angles add to $180^\circ$ . If you know one angle is $90^\circ$ , the other two must sum to $90^\circ$ , so they cannot both be $60^\circ$ !

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Triangle Properties Analysis: Can a 90° Right Triangle Be Equilateral?

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