The Pythagorean Theorem is a fundamental principle in geometry, particularly applicable to right triangles. It 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. The theorem can be expressed with the following formula:
\[ c^2 = a^2 + b^2 \]
Where:
- \( c \) is the length of the hypotenuse,
- \( a \) and \( b \) are the lengths of the other two sides.
How to Use the Pythagorean Theorem:
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Identify the sides: Determine which side is the hypotenuse and which are the other two sides.
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Plug the values into the equation: If you know the lengths of any two sides, substitute those values into the formula.
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Solve for the unknown side: Rearrange the equation to find the missing side:
- If finding the hypotenuse: \[ c = \sqrt{a^2 + b^2} \]
- If finding one of the other sides: \[ a = \sqrt{c^2 - b^2} \] or \( b = \sqrt{c^2 - a^2} \)
Example Problem:
Problem: A right triangle has legs of lengths 3 units and 4 units. What is the length of the hypotenuse?
Solution:
- Identify \( a = 3 \) and \( b = 4 \).
- Use the Pythagorean Theorem: \[ c^2 = a^2 + b^2 \] \[ c^2 = 3^2 + 4^2 \] \[ c^2 = 9 + 16 \] \[ c^2 = 25 \]
- Take the square root: \[ c = \sqrt{25} \] \[ c = 5 \]
So, the length of the hypotenuse is 5 units.
Application of the Pythagorean Theorem:
- Determining distances: It can be used to calculate the distance between two points in a Cartesian coordinate system.
- Real-world applications: Architecture, construction, navigation, and various fields of science.
Practice Problems:
- A right triangle has one leg measuring 6 units and the hypotenuse measuring 10 units. Find the length of the other leg.
- In a right triangle, the lengths of both legs are equal. If each leg is \( x \), what is the length of the hypotenuse in terms of \( x \)?
- A ladder is leaning against a wall. The foot of the ladder is 3 feet from the wall, and the ladder reaches 4 feet up the wall. Is this ladder set up correctly as a right triangle?
Feel free to ask for solutions to these practice problems or for more information on related concepts!
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