why is it a^2 - b^2=c^2 in pythgrium therium when its a right triangle

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Let's clarify a common misconception about the Pythagorean theorem. In a right triangle, we have two legs labeled 'a' and 'b', and the hypotenuse labeled 'c'. The correct Pythagorean theorem states that a squared plus b squared equals c squared, not a squared minus b squared equals c squared. The subtraction formula is mathematically incorrect for right triangles. To understand why we use addition in the Pythagorean theorem, let's visualize it with squares. We construct a square on each side of our right triangle. The square on side 'a' has area a squared, the square on side 'b' has area b squared, and the square on the hypotenuse 'c' has area c squared. The theorem states that the sum of the two smaller square areas equals the largest square area. For our 3-4-5 triangle: 3 squared plus 4 squared equals 9 plus 16, which equals 25, which is 5 squared. Here's the classic geometric proof using square rearrangement. We start with a large square of side length a plus b. This square contains four identical right triangles arranged around a central square. The total area of the large square is a plus b squared. This same area can also be calculated as four times one-half a b, plus c squared. Expanding the first expression gives a squared plus 2ab plus b squared. Setting the two expressions equal and simplifying, we get a squared plus b squared equals c squared. Now let's directly address why we use addition instead of subtraction. Using our 3-4-5 triangle, let's test both formulas. The correct formula gives us 3 squared plus 4 squared equals 9 plus 16, which equals 25, which is indeed 5 squared. This works perfectly. However, if we tried the incorrect subtraction formula, we would get 3 squared minus 4 squared equals 9 minus 16, which equals negative 7. This cannot equal 5 squared, which is 25. More importantly, negative areas are geometrically impossible, making the subtraction formula meaningless for real triangles. Let's see how the correct Pythagorean theorem works in practical situations. First, finding the diagonal of a rectangle with width 6 and height 8: the diagonal equals the square root of 6 squared plus 8 squared, which is the square root of 100, giving us 10 units. Second, calculating distance between coordinate points from origin to point 3,4: the distance is the square root of 3 squared plus 4 squared, which equals 5 units. Third, determining ladder length when the base is 5 meters from a wall and reaches 12 meters high: the ladder length is the square root of 5 squared plus 12 squared, which equals 13 meters. All these real-world applications give positive, meaningful results that match physical reality.