Using Pythagoras's Theorem to Calculate Missing Triangle Lengths

Key Points

Pythagoras's Theorem is essential for solving for missing lengths in right-angled triangles.
Equation: ( a^2 + b^2 = c^2 ).
Conditions for using the theorem:
1. The triangle must have a right angle (90 degrees).
2. You must know the lengths of two sides.
3. There must be one missing length to find.

Variables a, b, and c represent the sides of the triangle:
- ( c ) is the hypotenuse (longest side, opposite the right angle).
- ( a ) and ( b ) can be any of the other two sides.

Example 1:
- Known sides: ( a = 4, b = 3 ).
- Equation: ( 4^2 + 3^2 = c^2 ).
- Simplify: ( 16 + 9 = c^2 ), resulting in ( 25 = c^2 ).
- Solve: ( c = \sqrt{25} = 5 ).

Example 2:
- Known sides: ( a = 1.7, b = 3.2 ).
- Equation: ( 1.7^2 + 3.2^2 = x^2 ).
- Calculator: Result in ( 13.13 = x^2 ).
- Solve: ( x = \sqrt{13.13} \approx 3.62 ) (rounded to 3 significant figures).
Example 3:
- Known sides: ( a = 5.6, b = 10.5 ).
- Equation: ( 5.6^2 + 10.5^2 = c^2 ).
- Calculator: Result in ( 141.61 = c^2 ).
- Solve: ( c = \sqrt{141.61} = 11.9 ).
Example 4:
- Known sides: ( a = 8, b = 11 ).
- Equation: ( 8^2 + 11^2 = c^2 ).
- Simplify: ( 64 + 121 = c^2 ), resulting in ( 185 = c^2 ).
- Solve: ( c = \sqrt{185} \approx 13.6 ).

Labeling conventions: Often, exam questions label corners with letters. Use these to identify sides initially but follow the standard labeling of a, b, c for calculations.
Calculator Use: For accuracy, especially when dealing with decimals, it's advised to use a calculator.

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