Explain Pythagoras' law---**Textual Information:** * Title: Pythagorean Theorem * Mathematical Formula: c² = a² + b² * Labels on the Triangle: a, b, c **Chart/Diagram Description:** * Type: Geometric figure (Right-angled triangle) displayed on a blackboard. * Main Elements: * A right-angled triangle is shown with vertices at approximately the bottom left, top left, and bottom right of the triangle. * The vertical side is labeled 'a'. * The horizontal side is labeled 'b'. * The hypotenuse (diagonal side) is labeled 'c'. * A square symbol indicating a right angle is shown at the vertex between sides 'a' and 'b'. * The labels 'a', 'b', and 'c' are positioned next to the respective sides. * The mathematical formula "c² = a² + b²" is written to the right of the triangle. * The title "Pythagorean Theorem" is written above the triangle and formula. * Background: A green surface resembling a blackboard with a brown wooden frame. * Other Objects: Two white chalk pieces and a white object (possibly an eraser or brush) are visible at the bottom of the frame.

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Welcome to our exploration of the Pythagorean Theorem. This fundamental principle in geometry establishes a relationship between the sides of a right-angled triangle. The theorem states that in a right triangle, the square of the length of the hypotenuse, labeled c, is equal to the sum of the squares of the other two sides, labeled a and b. This is expressed by the formula: c squared equals a squared plus b squared. Let's visualize the Pythagorean Theorem. We can draw squares on each side of our right triangle. The square on side a has an area of a squared. The square on side b has an area of b squared. And the square on the hypotenuse c has an area of c squared. The theorem tells us that the sum of the areas of the squares on the two legs equals the area of the square on the hypotenuse. This gives us our formula: c squared equals a squared plus b squared. Let's verify the Pythagorean Theorem with some numerical examples. First, consider a 3-4-5 triangle. According to the theorem, 5 squared should equal 3 squared plus 4 squared. Computing this, we get 25 equals 9 plus 16, which is true. Another common example is the 5-12-13 triangle. Here, 13 squared should equal 5 squared plus 12 squared. Indeed, 169 equals 25 plus 144. These examples confirm that the Pythagorean Theorem works for right triangles with different side lengths. The Pythagorean Theorem has numerous real-world applications. One common use is in distance calculation. For example, if we have two points on a coordinate plane, we can find the distance between them using the Pythagorean Theorem. If point A is at coordinates (1,1) and point B is at (5,4), the horizontal distance is 4 units and the vertical distance is 3 units. Using the Pythagorean Theorem, the direct distance equals the square root of 4 squared plus 3 squared, which is the square root of 25, or exactly 5 units. This principle is also used in navigation, mapping, architecture, construction, engineering, and physics. To summarize what we've learned about the Pythagorean Theorem: First, it states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides, expressed as c squared equals a squared plus b squared. This theorem applies to all right triangles regardless of their size. We can visualize this relationship by drawing squares on each side of the triangle. Common examples that satisfy this theorem include the 3-4-5 and 5-12-13 triangles. Finally, the Pythagorean Theorem has numerous practical applications in fields like distance calculation, navigation, architecture, and engineering.