Function of the Day:
How many classifications (types) of triangles are there?
There are primarily three classifications of triangles based on their side lengths and angles:
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By Side Lengths:
- Equilateral Triangle: All three sides are equal.
- Isosceles Triangle: Two sides are equal in length.
- Scalene Triangle: All three sides are of different lengths.
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By Angle Measures:
- Acute Triangle: All three angles are less than 90 degrees.
- Right Triangle: One angle is exactly 90 degrees.
- Obtuse Triangle: One angle is greater than 90 degrees.
What do you know?
- Triangles can be classified by both side lengths and angles.
- The sum of the interior angles of a triangle is always 180 degrees.
- The Pythagorean theorem applies to right triangles: \(a^2 + b^2 = c^2\).
- An equilateral triangle is also a special case of an acute triangle.
- The altitude of a triangle can help you find its area.
- Isosceles triangles have at least two equal sides and angles.
- Triangles can be identified using their vertices (A, B, C).
- Triangles can be drawn on different coordinate planes.
- Triangles can be used in real-world applications like architecture and design.
- The concept of congruence can be applied to triangles.
Vocabulary Terms - Define/Illustrate
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Acute Angle: An angle less than 90 degrees.
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Obtuse Angle: An angle greater than 90 degrees but less than 180 degrees.
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Right Triangle: A triangle with one angle measuring 90 degrees.
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Altitude of a Triangle: The perpendicular segment from a vertex to the line containing the opposite side.
5 Problems - SHOW ALL WORK!
Problem 1: Right Triangles
Determine which of the following sets of side lengths can create a right triangle:
- b. (3, 4, 5)
- c. (5, 12, 13)
- d. (1, 1, 1)
- e. (7, 24, 25)
- f. (8, 15, 17)
Steps to Show Work:
- Use the Pythagorean theorem: If \(a^2 + b^2 = c^2\) (where \(c\) is the longest side), then it's a right triangle.
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For (3, 4, 5): \[3^2 + 4^2 = 9 + 16 = 25 = 5^2\] (Right triangle)
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For (5, 12, 13): \[5^2 + 12^2 = 25 + 144 = 169 = 13^2\] (Right triangle)
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For (1, 1, 1): \[1^2 + 1^2 \neq 1^2\] (Not a right triangle)
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For (7, 24, 25): \[7^2 + 24^2 = 49 + 576 = 625 = 25^2\] (Right triangle)
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For (8, 15, 17): \[8^2 + 15^2 = 64 + 225 = 289 = 17^2\] (Right triangle)
Right Triangles: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17).
Problem 2: Solve the equations with exponents.
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\(x^2 + 5 = 20\)
- Subtract 5 from both sides: \[x^2 = 15\]
- Take square root: \[x = \sqrt{15} \text{ or } x = -\sqrt{15}\]
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\(m^3/3 - 2 = 22\)
- Add 2 to both sides: \[m^3/3 = 24\]
- Multiply by 3: \[m^3 = 72\]
- Take the cube root: \[m = \sqrt[3]{72} = 4.16 \text{ (approximately)}\]
Problem 3: Visual Representation of 100
The number 100 can be visually represented as a square with sides of length 10:
- Area: \(10 \times 10 = 100\)
This representation signifies a complete set or a whole number, often used in contexts like scoring, measurements, or percentages.
Problem 4: Area of a Triangle
The area of a triangle is given by: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
This relates to a rectangle or square since if you take a rectangle with the same base and height, its area is: \[ \text{Area of Rectangle} = \text{base} \times \text{height} \]
The area of a triangle is half that of a rectangle with the same base and height.
Problem 5: Relation of Squares and Square Roots to Triangles
Squares and square roots relate to triangles in the context of the Pythagorean theorem, which involves squaring the lengths of the sides of right triangles. The theorem helps determine unknown lengths (the square roots).
What do I need help with?
Please specify the areas or topics within the triangle study and math concepts that you feel need further explanation or examples!