Finding the Hypotenuse Using the Pythagorean Theorem

Introduction

• Objective: Learn how to find the length of the hypotenuse in a right triangle using the Pythagorean theorem.
• Pythagorean theorem: (a^2 + b^2 = c^2)
  • (c): Hypotenuse (longest side, opposite the right angle)
  • (a) and (b): Legs of the triangle

Example 1

• Given: Right triangle with legs 8 meters and 6 meters.
• Find: Length of the hypotenuse ((c)).
• Process:
  1. Identify (a = 8) and (b = 6).
  2. Plug into the equation: [8^2 + 6^2 = c^2]
  3. Calculate:
     • (8^2 = 64)
     • (6^2 = 36)
     • (64 + 36 = 100)
  4. Solve for (c):
     • (c^2 = 100)
     • (c = \sqrt{100} = 10)
• Result: Hypotenuse (c = 10) meters.

Example 2

• Given: Right triangle with legs 10 feet and 7 feet.
• Find: Length of the hypotenuse ((c)).
• Process:
  1. Identify (a = 10) and (b = 7).
  2. Plug into the equation: [10^2 + 7^2 = c^2]
  3. Calculate:
     • (10^2 = 100)
     • (7^2 = 49)
     • (100 + 49 = 149)
  4. Solve for (c):
     • (c^2 = 149)
     • (c = \sqrt{149})
     • (\sqrt{149}) is approximately 12.21 when rounded to the hundredths place.
• Result: Hypotenuse (c \approx 12.21) feet.

Key Points

• Hypotenuse: Longest side of a right triangle, opposite the right angle.
• Calculation: Use the Pythagorean theorem to solve for the hypotenuse if two sides of a right triangle are known.
• Square Roots: Beware of non-perfect squares; use approximate values for irrational numbers.

Conclusion

• Application: Use these steps and examples to find the hypotenuse in any right triangle using the Pythagorean theorem.