Using Pythagoras's Theorem to Calculate Missing Triangle Lengths

When to Use Pythagoras's Theorem

• Right Angle Triangle: The triangle must have a 90-degree angle.
• Two Known Sides: You need to know the lengths of any two sides.
• One Missing Side: There should be one side whose length is unknown.

Pythagoras's Theorem Formula

• Equation: ( a^2 + b^2 = c^2 )
  • Memorize this equation as it is frequently used.
• Definitions:
  • ( c ) is the hypotenuse (longest side, opposite the right angle).
  • ( a ) and ( b ) are the other two sides (can be interchanged).

Example Calculations

Example 1

1. Triangle Given: Known sides are 3 and 4.
2. Label Sides: Assign ( a = 4 ), ( b = 3 ), ( c = x ).
3. Solve Equation:
   • ( 4^2 + 3^2 = c^2 )
   • Simplifies to ( 16 + 9 = c^2 )
   • ( 25 = c^2 )
4. Find ( c ):
   • Square root both sides: ( c = 5 )

Example 2

1. Triangle Given: Known sides are 1.7 and 3.2.
2. Label Sides: Unknown side is ( c ), longest side.
3. Write Equation:
   • ( 1.7^2 + 3.2^2 = c^2 )
   • Use a calculator for ( 1.7^2 + 3.2^2 = 13.13 )
4. Solve for ( x ):
   • Square root both sides: ( x = 3.62 )

Example 3

• Labeling and Solving Steps:
  1. Ignore corner labels when using Pythagoras.
  2. Assign sides a, b, c according to the triangle.
  3. Write and solve equation as demonstrated.

Important Tips

• Corners Labeling: Exam questions may use corner labels (e.g., X, Y, Z) to indicate sides.
  • Example: Find length ( XZ ) means calculate side between X and Z.
• Final Step: Always square root to find the length of the missing side.

Conclusion

• Applying the Pythagoras theorem allows you to solve for missing side lengths in right-angle triangles efficiently.
• Practice problems can enhance understanding and proficiency.