Pythagorean Theorem Overview

Example 1: Legs of 4 feet and 3 feet.
- Calculate using ( 4^2 + 3^2 = c^2 ).
- Results in ( 16 + 9 = 25 ), thus ( c = 5 ) feet.
Visual Representation:
- Areas of squares on the legs equal the area of the square on the hypotenuse: ( 16 ) sq ft + ( 9 ) sq ft = ( 25 ) sq ft.

Example 1: Given legs of 8 meters and 6 meters.
- Calculation: ( 8^2 + 6^2 = c^2 ).
- Results in ( 64 + 36 = 100 ), thus ( c = 10 ) meters.
Example 2: Given legs of 10 feet and 7 feet.
- Calculation: ( 10^2 + 7^2 = c^2 ).
- Results in ( 100 + 49 = 149 ), thus ( c \approx 12.21 ) feet (rounded).

Example 1: Given leg of 5 yards and hypotenuse of 13 yards.
- Setup as ( a^2 + 5^2 = 13^2 ).
- Results in ( a^2 + 25 = 169 ), thus ( a = 12 ) yards.
Example 2: Given leg of 4 meters and hypotenuse of 9 meters.
- Setup as ( 4^2 + b^2 = 9^2 ).
- Results in ( 16 + b^2 = 81 ), thus ( b \approx 8.06 ) meters (rounded).

In right triangles with decimal side lengths, use the same Pythagorean theorem formula ( a^2 + b^2 = c^2 ).
Important to calculate squares accurately and keep track of decimal places for precision.

The Pythagorean theorem is a fundamental principle in geometry used to find the relationship between the sides of right triangles.
Mastery of the theorem enables calculations of missing side lengths, given certain known sides, applying consistently across various scenarios and calculations.### Pythagorean Theorem Overview
The theorem states that in a right triangle, the square of the hypotenuse (C) equals the sum of the squares of the other two sides (A and B).
Represented by the equation: (C^2 = A^2 + B^2).

Given:
- A = 44.89
- B = 231.04
Calculation:
- Sum A and B: (44.89 + 231.04 = 275.93).
- Isolate C: (C = \sqrt{275.93}).
Result:
- The square root of 275.93 is approximately 16.61, which rounds to 16.6 feet.

Given:
- A = 8.3 meters
- C (hypotenuse) = 12.6 meters.
Calculation:
- Use the formula: (A^2 + B^2 = C^2).
- Calculate squares: (8.3^2 = 68.89) and (12.6^2 = 158.76).
- Subtract: (B^2 = 158.76 - 68.89 = 89.87).
Result:
- Isolate B: (B = \sqrt{89.87}).
- The square root of 89.87 is approximately 9.54, rounding to 9.5 meters.

C represents the hypotenuse while A and B represent the triangle's legs.
Square roots of non-perfect squares yield irrational numbers, often rounded for practical use.
Measurements given in feet (for the hypotenuse) and meters (for the missing leg) are consistent with the problem context.

The theorem defines a fundamental relationship in right triangles, where the square of the hypotenuse equals the sum of the squares of the other two sides.
Named after Pythagoras, significant in mathematics and philosophy.
Applicable solely to right triangles.

Expressed mathematically as ( a^2 + b^2 = c^2 ).
( a ) and ( b ) represent the lengths of the legs while ( c ) is the hypotenuse.
Enables calculation of a side length when the other two lengths are known.

Example 1: For legs measuring 4 feet and 3 feet, ( 4^2 + 3^2 = c^2 ) results in ( c = 5 ) feet.
Visual Representation: Areas of squares on the legs equal the area of the square on the hypotenuse, confirming ( 16 + 9 = 25 ).

Example 1: Legs of 8 meters and 6 meters calculate to ( c = 10 ) meters.
Example 2: Legs of 10 feet and 7 feet result in ( c \approx 12.21 ) feet when solved.

Example 1: A leg of 5 yards with a hypotenuse of 13 yards results in the other leg measuring 12 yards.
Example 2: A leg of 4 meters with a hypotenuse of 9 meters gives a missing leg of approximately 8.06 meters.

The theorem applies to triangles with decimal side lengths, requiring careful square calculations and attention to decimal placement for precision.

Mastering the Pythagorean theorem is crucial for accurately determining unknown side lengths in right triangles across various applications.

To find the hypotenuse given ( A = 44.89 ) and ( B = 231.04 ), isolate ( C ) to approximate 16.6 feet.
For a missing leg with ( A = 8.3 ) meters and ( C = 12.6 ) meters, the result for ( B ) approximates to 9.5 meters.

( C ) signifies the hypotenuse, while ( A ) and ( B ) denote the legs.
Square roots of non-perfect squares typically result in irrational numbers, usually rounded for practical purposes.
Consistent measurement units enhance clarity in problem-solving, using feet for hypotenuse and meters for legs.