Question
pythagoren therom portfolio worksheet
Answers
GPT-4o mini
Creating a portfolio worksheet on the Pythagorean theorem can be a fun and educational way to explore this fundamental concept of geometry. Below is a suggested outline for your portfolio worksheet, including various sections, examples, and activities.
### Pythagorean Theorem Portfolio Worksheet
---
**Name:** ______________________
**Date:** ______________________
---
### Section 1: Introduction to the Pythagorean Theorem
**Definition:**
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as:
\[ c^2 = a^2 + b^2 \]
where:
- \( c \) = length of the hypotenuse
- \( a \) and \( b \) = lengths of the other two sides
---
### Section 2: Diagram of a Right Triangle
**Draw a diagram of a right triangle and label the sides.**
- Label the hypotenuse as \( c \) and the other two sides as \( a \) and \( b \).
**Diagram:**
(Provide space for a diagram or drawings)
---
### Section 3: Examples
**Example 1: Basic Calculation**
For a right triangle where \( a = 3 \) and \( b = 4 \):
1. Use the Pythagorean theorem to find \( c \).
\[ c^2 = 3^2 + 4^2 \]
\[ c^2 = 9 + 16 \]
\[ c^2 = 25 \]
\[ c = 5 \]
---
**Example 2: Finding a Side**
Given \( c = 10 \) and \( a = 6 \), find \( b \):
1. Rearrange the Pythagorean theorem:
\[ b^2 = c^2 - a^2 \]
2. Plug in the values:
\[ b^2 = 10^2 - 6^2 \]
\[ b^2 = 100 - 36 \]
\[ b^2 = 64 \]
\[ b = 8 \]
---
### Section 4: Application Problems
**Problem 1: Real-Life Scenario**
A ladder is leaning against a wall. The base of the ladder is 4 feet from the wall, and the top of the ladder reaches 3 feet high on the wall. How long is the ladder?
**Solution:**
Using the Pythagorean theorem:
- Let \( a = 3 \), \( b = 4 \).
- Find \( c \).
**Answer:**
\[ c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ feet} \]
---
**Problem 2: Finding Distance**
Two points A(1, 2) and B(4, 6) are on a plane. Find the distance between points A and B using the Pythagorean theorem.
1. Calculate the lengths of sides \( a \) and \( b \):
- \( a = 4 - 1 = 3 \)
- \( b = 6 - 2 = 4 \)
2. Use the Pythagorean theorem to find \( c \).
**Answer:**
\[ c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
---
### Section 5: Practice Problems
1. If \( a = 5 \) and \( b = 12 \), find \( c \).
2. If \( c = 13 \) and \( b = 5 \), find \( a \).
3. A right triangle has a hypotenuse of 15 feet and one side of 9 feet. Find the length of the other side.
---
### Section 6: Reflection
**What did you learn about the Pythagorean theorem?**
(Provide space for students to write their reflections)
---
### Section 7: Additional Resources
- **Books:** List any relevant textbooks or resources.
- **Videos:** Suggest online videos explaining the Pythagorean theorem.
- **Websites:** Include educational websites for further practice (e.g., Khan Academy, Math is Fun).
---
Feel free to customize this worksheet according to your needs or the intended audience!
### Pythagorean Theorem Portfolio Worksheet
---
**Name:** ______________________
**Date:** ______________________
---
### Section 1: Introduction to the Pythagorean Theorem
**Definition:**
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as:
\[ c^2 = a^2 + b^2 \]
where:
- \( c \) = length of the hypotenuse
- \( a \) and \( b \) = lengths of the other two sides
---
### Section 2: Diagram of a Right Triangle
**Draw a diagram of a right triangle and label the sides.**
- Label the hypotenuse as \( c \) and the other two sides as \( a \) and \( b \).
**Diagram:**
(Provide space for a diagram or drawings)
---
### Section 3: Examples
**Example 1: Basic Calculation**
For a right triangle where \( a = 3 \) and \( b = 4 \):
1. Use the Pythagorean theorem to find \( c \).
\[ c^2 = 3^2 + 4^2 \]
\[ c^2 = 9 + 16 \]
\[ c^2 = 25 \]
\[ c = 5 \]
---
**Example 2: Finding a Side**
Given \( c = 10 \) and \( a = 6 \), find \( b \):
1. Rearrange the Pythagorean theorem:
\[ b^2 = c^2 - a^2 \]
2. Plug in the values:
\[ b^2 = 10^2 - 6^2 \]
\[ b^2 = 100 - 36 \]
\[ b^2 = 64 \]
\[ b = 8 \]
---
### Section 4: Application Problems
**Problem 1: Real-Life Scenario**
A ladder is leaning against a wall. The base of the ladder is 4 feet from the wall, and the top of the ladder reaches 3 feet high on the wall. How long is the ladder?
**Solution:**
Using the Pythagorean theorem:
- Let \( a = 3 \), \( b = 4 \).
- Find \( c \).
**Answer:**
\[ c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ feet} \]
---
**Problem 2: Finding Distance**
Two points A(1, 2) and B(4, 6) are on a plane. Find the distance between points A and B using the Pythagorean theorem.
1. Calculate the lengths of sides \( a \) and \( b \):
- \( a = 4 - 1 = 3 \)
- \( b = 6 - 2 = 4 \)
2. Use the Pythagorean theorem to find \( c \).
**Answer:**
\[ c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
---
### Section 5: Practice Problems
1. If \( a = 5 \) and \( b = 12 \), find \( c \).
2. If \( c = 13 \) and \( b = 5 \), find \( a \).
3. A right triangle has a hypotenuse of 15 feet and one side of 9 feet. Find the length of the other side.
---
### Section 6: Reflection
**What did you learn about the Pythagorean theorem?**
(Provide space for students to write their reflections)
---
### Section 7: Additional Resources
- **Books:** List any relevant textbooks or resources.
- **Videos:** Suggest online videos explaining the Pythagorean theorem.
- **Websites:** Include educational websites for further practice (e.g., Khan Academy, Math is Fun).
---
Feel free to customize this worksheet according to your needs or the intended audience!