122 lines
4.7 KiB
Markdown
122 lines
4.7 KiB
Markdown
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# explain_math
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explain_math is a prompt designed to help you understand mathematical concepts in a clear and engaging way.
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## Example:
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This example was run on llama3:latest (8B), even with the 8B limitation, the example is still clear and engaging.
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<details open>
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<summary>Raw</summary>
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```markdown
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**Pythagorean Theorem: Understanding the Relationship Between Triangles and Squares**
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The Pythagorean theorem, named after ancient Greek philosopher and mathematician Pythagoras, is a fundamental concept in geometry that helps us calculate the length of the hypotenuse (the longest side) of a right-angled triangle. This theorem has far-reaching applications in various fields, including mathematics, physics, engineering, and architecture.
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**The Formula:**
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a^2 + b^2 = c^2
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where:
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* `c` is the length of the hypotenuse (the longest side)
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* `a` and `b` are the lengths of the other two sides (the legs)
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**Step-by-Step Explanation:**
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Let's consider a right-angled triangle with one angle that is 90 degrees (a square corner). We can use this theorem to find the length of the hypotenuse (`c`) when we know the lengths of the other two sides (`a` and `b`).
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1. Square both legs (`a` and `b`) by multiplying them by themselves.
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2. Add the squared values of the legs together: a^2 + b^2
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3. The result is equal to the square of the hypotenuse (`c`): c^2
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**Example:**
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Suppose we have a right-angled triangle with one leg that is 3 inches long and the other leg is 4 inches long. We want to find the length of the hypotenuse.
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1. Square both legs: 3^2 = 9, 4^2 = 16
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2. Add the squared values together: 9 + 16 = 25
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3. The result is equal to the square of the hypotenuse: c^2 = 25
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Now, take the square root of both sides:
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c = √25 ≈ 5 inches
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**Real-World Applications:**
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The Pythagorean theorem has numerous practical applications in various fields, including:
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* Construction and architecture: Finding the length of diagonals, beams, or pillars.
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* Physics: Calculating distances, velocities, and accelerations in problems involving right triangles.
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* Engineering: Designing bridges, buildings, or other structures that rely on triangular shapes.
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**Visual Aids:**
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For a better understanding, you can use online tools or software to create interactive diagrams that demonstrate the Pythagorean theorem. Some popular options include:
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* GeoGebra
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* Desmos
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* Khan Academy's math exercises
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Remember, this theorem is not just a mathematical concept; it has many practical applications in various fields.
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```
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</details>
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<details>
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<summary>Rendered</summary>
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**Pythagorean Theorem: Understanding the Relationship Between Triangles and Squares**
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|
|
|
||
|
|
The Pythagorean theorem, named after ancient Greek philosopher and mathematician Pythagoras, is a fundamental concept in geometry that helps us calculate the length of the hypotenuse (the longest side) of a right-angled triangle. This theorem has far-reaching applications in various fields, including mathematics, physics, engineering, and architecture.
|
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|
|
|
||
|
|
**The Formula:**
|
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|
|
|
||
|
|
a^2 + b^2 = c^2
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|
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|
||
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|
where:
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|
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* `c` is the length of the hypotenuse (the longest side)
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* `a` and `b` are the lengths of the other two sides (the legs)
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|
|
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||
|
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**Step-by-Step Explanation:**
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|
|
|
||
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|
Let's consider a right-angled triangle with one angle that is 90 degrees (a square corner). We can use this theorem to find the length of the hypotenuse (`c`) when we know the lengths of the other two sides (`a` and `b`).
|
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|
|
|
||
|
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1. Square both legs (`a` and `b`) by multiplying them by themselves.
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2. Add the squared values of the legs together: a^2 + b^2
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3. The result is equal to the square of the hypotenuse (`c`): c^2
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**Example:**
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Suppose we have a right-angled triangle with one leg that is 3 inches long and the other leg is 4 inches long. We want to find the length of the hypotenuse.
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|
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1. Square both legs: 3^2 = 9, 4^2 = 16
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2. Add the squared values together: 9 + 16 = 25
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3. The result is equal to the square of the hypotenuse: c^2 = 25
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Now, take the square root of both sides:
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c = √25 ≈ 5 inches
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**Real-World Applications:**
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The Pythagorean theorem has numerous practical applications in various fields, including:
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|
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* Construction and architecture: Finding the length of diagonals, beams, or pillars.
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* Physics: Calculating distances, velocities, and accelerations in problems involving right triangles.
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* Engineering: Designing bridges, buildings, or other structures that rely on triangular shapes.
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|
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**Visual Aids:**
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|
||
|
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For a better understanding, you can use online tools or software to create interactive diagrams that demonstrate the Pythagorean theorem. Some popular options include:
|
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|
|
|
||
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* GeoGebra
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* Desmos
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* Khan Academy's math exercises
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Remember, this theorem is not just a mathematical concept; it has many practical applications in various fields.
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</details>
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