Using Pythagoras' Theorem to Calculate the Missing Length of a Triangle

Key Criteria for Using Pythagoras' Theorem

• Right-angled triangle: The triangle must have a 90-degree angle.
• Two known side lengths: You need to know the lengths of any two sides.
• One missing length: There should be one side whose length you need to find (e.g., x).

Pythagoras' Theorem Formula

• The equation: a² + b² = c²
• c: Always the hypotenuse (the longest side, opposite the right angle).
• a and b: The other two sides, placement doesn't matter.

Example Calculation Steps

Example 1: Known sides are 3 and 4; find hypotenuse (c).

• Label sides: 3 = a, 4 = b
• Apply formula:
  • 4² + 3² = c²
  • 16 + 9 = c²
  • 25 = c²
• Solve for c:
  • √25 = c
  • c = 5

Example 2: Known sides are 1.7 and 3.2; find hypotenuse (c or x).

• Label sides: 1.7 = a, 3.2 = b
• Apply formula:
  • 1.7² + 3.2² = x²
  • Using calculator: 2.89 + 10.24 = 13.13 = x²
• Solve for x:
  • √13.13 = x
  • x ≈ 3.62 (to three significant figures)

Example 3: Given sides are 5.6 and 10.5; find hypotenuse (xz).

• Label sides: 5.6 = a, 10.5 = b
• Apply formula:
  • 5.6² + 10.5² = c²
  • Using calculator: 31.36 + 110.25 = 141.61 = c²
• Solve for c:
  • √141.61 = c
  • c ≈ 11.9

Example 4: Known sides are 8 and 11; find hypotenuse (ac).

• Label sides: 8 = a, 11 = b
• Apply formula:
  • 8² + 11² = c²
  • Using calculator: 64 + 121 = 185 = c²
• Solve for c:
  • √185 = c
  • c ≈ 13.6

Notes on Notation

• Exam questions may label corners and provide line segments (e.g., xz, ac) to specify the missing side.
• Ignore the specific labels (x, y, z, etc.) during calculation; follow standard a, b, c labeling for Pythagoras' theorem.

Conclusion

• Always check that the triangle is right-angled.
• Ensure you know two side lengths before applying the theorem.
• Plug in the values and solve algebraically, using a calculator where necessary for squares and square roots.

Remember to review the content and practice with more problems!