Solving for X in a Right Triangle Using the Pythagorean Theorem

Key Concepts

• Pythagorean Theorem: In a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse.

  [ a^2 + b^2 = c^2 ]

  Where:

  • (a) and (b) are the lengths of the legs.
  • (c) is the length of the hypotenuse.

Example Problem

• Given: A right triangle with leg lengths of 6 and 8, and an unknown hypotenuse (x).

Steps to Solve

1. Set Up the Equation:

   • According to the Pythagorean theorem:

     [ 6^2 + 8^2 = x^2 ]

2. Simplify Each Term:

   • Calculate (6^2):
     • (6 \times 6 = 36)
   • Calculate (8^2):
     • (8 \times 8 = 64)

3. Combine and Simplify:

   • Sum the squares:

     • (36 + 64 = 100)

   • So, the equation becomes:

     [ 100 = x^2 ]

4. Solve for (x):

   • Take the square root of both sides:
     • (\sqrt{100} = 10)
     • (\sqrt{x^2} = x)
   • Thus, (x = 10)

Conclusion

• The value of (x), representing the hypotenuse, is 10.
• Note: Only the positive root is considered since the length cannot be negative.