Classification of Triangles
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Classification of Triangles by Sides
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Today, we’re diving into the classification of triangles! Let’s start by discussing how we can categorize triangles based on the lengths of their sides. Can anyone tell me what a scalene triangle is?
Is it a triangle where all three sides have different lengths?
Exactly, great job! A scalene triangle has all sides unequal. Now, who can tell me about an isosceles triangle?
That’s a triangle with two sides that are equal.
Correct! And what about an equilateral triangle?
All three sides are equal!
Well done! Remember, you can use the acronym 'SEE' to help remember this: S for scalene, E for equilateral, and E for isosceles.
That’s a helpful way to memorize them!
Exactly! Now, let’s summarize what we learned about triangles classified by sides: scalene has all sides different, isosceles has two equal, and equilateral has all equal.
Classification of Triangles by Angles
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Now, let’s shift gears and discuss how triangles can be classified based on their angles. Who remembers how we classify triangles by angles?
I think there are acute, right, and obtuse triangles?
That’s right! An acute triangle has all angles less than 90°.
What about a right triangle?
Good question! A right triangle has one angle that is exactly 90°. And what can you tell me about an obtuse triangle?
It has one angle that is greater than 90°.
Fantastic! You all are doing great. To remember these properties, think 'A for acute, R for right, and O for obtuse.' How about we practice these in different triangles?
That sounds like fun! Let's do it!
Awesome! To summarize, we have acute (all angles < 90°), right (one angle = 90°), and obtuse (one angle > 90°).
Triangle Inequality Theorem
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Let’s now discuss the triangle inequality theorem. Can someone explain what it says?
It says that the sum of any two sides of a triangle must be greater than the third side.
Excellent! This is crucial because it lets us determine whether three lengths can actually form a triangle. Can anyone recall a situation when this theorem might fail?
If I have sides of lengths 3, 4, and 7, they wouldn’t form a triangle, right?
Precisely! Because 3 + 4 equals 7, you get a degenerate triangle — which doesn’t enclose space like a typical triangle.A degenerate triangle is a triangle where the three vertices are collinear, meaning they all lie on a single straight line. This results in the area of the triangle being zero. As you stated, it is a triangle in which the sum of the lengths of two sides equals the length of the third side. Always remember to check that.
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Classification by Sides
Chapter 1 of 1
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Chapter Content
By sides
• Scalene: all sides unequal
• Isosceles: two sides equal
• Equilateral: all three sides equal
Detailed Explanation
Triangles can be classified based on the lengths of their sides. Here are the three types:
1. Scalene Triangle: All three sides have different lengths. This means that no two sides are the same.
2. Isosceles Triangle: This type has at least two sides of equal length. The equal sides are often referred to as the legs, and the angle opposite these sides is called the vertex angle.
3. Equilateral Triangle: In this triangle, all three sides are equal in length, and consequently, all three angles are also equal, each measuring 60 degrees.
Examples & Analogies
Think of a sports team jersey. If you have jerseys of three different players (like different lengths of sides) and one player's jersey is shorter than the others, that represents a scalene triangle. If you have jerseys of two players that are the same size, that's an isosceles triangle. If all jerseys are the same size, that symbolizes an equilateral triangle.
Key Concepts
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Scalene Triangle: All sides unequal.
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Isosceles Triangle: Two sides equal.
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Equilateral Triangle: All sides equal.
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Acute Triangle: All angles <90°.
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Right Triangle: One angle =90°.
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Obtuse Triangle: One angle >90°.
Examples & Applications
A triangle with sides 4, 5, and 6 is scalene.
A triangle with angles of 30°, 60° is an acute triangle.An acute triangle has angles like 50°, 60°, and 70° — all less than 90° and 90° is a right triangle.
Memory Aids
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Rhymes
In a triangle so fine, three types we will find. Scalene's sides aren't the same, Isosceles has a twin flame. Equilateral's all aligned!
Stories
Once upon a time, there were three friends: Scalene, Isosceles, and Equilateral. Scalene never played with the same lengths, Isosceles always matched with one friend, and Equilateral loved to keep things equal. They dreamt of forming triangles together in their geometrical land.
Memory Tools
Remember 'SIE AOR' - Scalene, Isosceles, Equilateral (sides); Acute, Obtuse, Right (angles).
Acronyms
To remember the triangle types
S.I.E.A - S for Scalene
for Isosceles
for Equilateral
for Acute
for Obtuse
for Right.
Flash Cards
Glossary
- Scalene Triangle
A triangle with all sides of different lengths.
- Isosceles Triangle
A triangle with two sides of equal length.
- Equilateral Triangle
A triangle with all three sides of equal length.
- Acute Triangle
A triangle with all angles less than 90°.
- Right Triangle
A triangle with one angle that measures exactly 90°.
- Obtuse Triangle
A triangle with one angle greater than 90°.
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