Solving the Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in mathematics that relates the sides of a right triangle. It states that in a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.
What is the Pythagorean Theorem?
The Pythagorean Theorem is expressed as:
\[ a^2 + b^2 = c^2 \] where:
- a = One leg of the right triangle
- b = The other leg of the right triangle
- c = The hypotenuse (longest side)
Steps to Solve the Pythagorean Theorem
- Identify the two known sides of the right triangle.
- Substitute the values into the equation: \( a^2 + b^2 = c^2 \).
- Solve for the missing side by taking the square root if necessary.
Example 1: Finding the Hypotenuse
Given a right triangle with legs \( a = 3 \) and \( b = 4 \), find \( c \):
\[ 3^2 + 4^2 = c^2 \] \[ 9 + 16 = c^2 \] \[ 25 = c^2 \] \[ c = \sqrt{25} = 5 \]
Answer: The hypotenuse is 5.
Example 2: Finding a Missing Leg
Given a right triangle where \( c = 10 \) and \( a = 6 \), find \( b \):
\[ 6^2 + b^2 = 10^2 \] \[ 36 + b^2 = 100 \] \[ b^2 = 100 - 36 \] \[ b^2 = 64 \] \[ b = \sqrt{64} = 8 \]
Answer: The missing leg is 8.
Real-World Applications of the Pythagorean Theorem
This theorem is widely used in various fields such as:
- Construction and architecture (measuring distances and angles).
- Navigation and GPS calculations.
- Computer graphics and design.
- Physics and engineering applications.
Common Considerations When Using the Pythagorean Theorem
Only for Right Triangles: The Pythagorean Theorem applies only to right-angled triangles.
Checking Solutions: Always verify calculations by substituting values back into the equation.
Approximations: When working with non-perfect squares, approximate the square root value.
Example
Solving the Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in geometry that defines the relationship between the three sides of a right-angled triangle.
The theorem states that:
Pythagorean Theorem Formula
For a right-angled triangle:
\[ a^2 + b^2 = c^2 \]Where:
- a and b are the lengths of the two shorter sides (legs) of the triangle.
- c is the length of the hypotenuse (the longest side, opposite the right angle).
Steps to Solve the Pythagorean Theorem
- Step 1: Identify the given values (a, b, or c).
- Step 2: Substitute the known values into the formula.
- Step 3: Solve for the missing side by performing addition or subtraction.
- Step 4: Take the square root if solving for the hypotenuse.
Example 1: Finding the Hypotenuse
If a right-angled triangle has legs of length 3 and 4, find the hypotenuse.
\[ c^2 = 3^2 + 4^2 \] \[ c^2 = 9 + 16 = 25 \] \[ c = \sqrt{25} = 5 \]So, the hypotenuse is 5.
Example 2: Finding a Missing Leg
If the hypotenuse is 10 and one leg is 6, find the missing leg.
\[ 6^2 + b^2 = 10^2 \] \[ 36 + b^2 = 100 \] \[ b^2 = 100 - 36 = 64 \] \[ b = \sqrt{64} = 8 \]So, the missing leg is 8.
Real-Life Applications of the Pythagorean Theorem
The theorem is widely used in various fields such as:
- Construction and architecture for building stable structures.
- Navigation and GPS systems to calculate distances.
- Physics and engineering for solving problems related to forces and distances.
| Triangle | Theorem Formula | Calculation Steps | Solution |
|---|---|---|---|
| Right Triangle with legs 3 and 4 | \( c^2 = a^2 + b^2 \) | \( c^2 = 3^2 + 4^2 = 9 + 16 = 25 \) | \( c = \sqrt{25} = 5 \) |
| Right Triangle with legs 5 and 12 | \( c^2 = a^2 + b^2 \) | \( c^2 = 5^2 + 12^2 = 25 + 144 = 169 \) | \( c = \sqrt{169} = 13 \) |
| Right Triangle with legs 7 and 24 | \( c^2 = a^2 + b^2 \) | \( c^2 = 7^2 + 24^2 = 49 + 576 = 625 \) | \( c = \sqrt{625} = 25 \) |
| Right Triangle with legs 8 and 15 | \( c^2 = a^2 + b^2 \) | \( c^2 = 8^2 + 15^2 = 64 + 225 = 289 \) | \( c = \sqrt{289} = 17 \) |
| Right Triangle with legs 9 and 40 | \( c^2 = a^2 + b^2 \) | \( c^2 = 9^2 + 40^2 = 81 + 1600 = 1681 \) | \( c = \sqrt{1681} = 41 \) |
