{
"appName": "VisualMath AI Export",
"exportedAt": "2025-11-24T20:58:36.471Z",
"input": {
"mode": "text",
"text": "explain pythagoras theroem"
},
"concept": {
"conceptTitle": "Pythagorean Theorem",
"educationalGoal": "To visually understand and prove the relationship between the sides of a right-angled triangle: a² + b² = c².",
"explanation": "The Pythagorean Theorem is a fundamental principle in geometry that describes the relationship between the three sides of a **right-angled triangle**. A right-angled triangle is a triangle with one angle measuring 90 degrees.\n\nLet's label the two shorter sides (legs) as `a` and `b`, and the longest side (hypotenuse), which is opposite the right angle, as `c`. The theorem states that the square of the hypotenuse (`c²`) is equal to the sum of the squares of the other two sides (`a² + b²`).\n\nWe'll explore a classic visual proof that demonstrates this relationship by rearranging shapes!",
"steps": [
{
"stepTitle": "1. Introducing the Right Triangle",
"instruction": "Observe the right-angled triangle. Its two shorter sides are labeled `a` and `b`, and the longest side, the hypotenuse, is labeled `c`. The square symbol marks the 90-degree angle.",
"visualFocus": "A single right-angled triangle with sides a, b, c labeled, and the right angle marked with a square."
},
{
"stepTitle": "2. Squares on the Sides",
"instruction": "Drag the slider to visualize the squares built on each side of the triangle. The area of each square corresponds to a², b², and c² respectively.",
"visualFocus": "The right triangle, with squares built outwards from each side (a, b, and c). The areas a², b², c² are displayed next to their respective squares."
},
{
"stepTitle": "3. Proof Setup: First Arrangement",
"instruction": "To prove a² + b² = c², we'll use a larger square with side length (a+b). Observe how four copies of our original right triangle are arranged inside this large square. They form a smaller square in the center.",
"visualFocus": "A large square (side a+b). Inside, four identical right triangles are arranged, forming a central square with side c. The central square and the four triangles should be distinct."
},
{
"stepTitle": "4. Area Calculation: First Arrangement",
"instruction": "The total area of the large square can be expressed as the sum of its internal parts: the central square and the four triangles. Click 'Next' to see the calculation.",
"visualFocus": "Text: 'Area_large = Area(central square) + Area(4 triangles)'. Then, 'Area_large = c² + 4 * (1/2 * a * b)' simplifies to 'Area_large = c² + 2ab'. The central square and triangles are highlighted during calculation."
},
{
"stepTitle": "5. Proof Setup: Second Arrangement",
"instruction": "Now, let's take the *exact same four triangles* and arrange them differently within a second identical large square (also with side a+b). This time, they form two smaller squares (a² and b²) and two rectangles.",
"visualFocus": "A second large square (side a+b) appears next to the first. Inside it, the four identical right triangles are rearranged to form a square of side 'a', a square of side 'b', and two rectangles of area 'ab' each. The a² and b² squares should be distinct."
},
{
"stepTitle": "6. Area Calculation: Second Arrangement",
"instruction": "Similar to before, the total area of this second large square can also be expressed as the sum of *its* internal parts. Click 'Next' to see the calculation.",
"visualFocus": "Text: 'Area_large = Area(square a) + Area(square b) + Area(2 rectangles)'. Then, 'Area_large = a² + b² + 2 * (a * b)' is displayed. The a² and b² squares and the 'ab' rectangles are highlighted during calculation."
},
{
"stepTitle": "7. The Conclusion",
"instruction": "Since both large squares have the same side length (a+b) and contain the exact same four triangles, their total areas must be equal. Equate the two expressions for the large square's area to reveal the Pythagorean Theorem!",
"visualFocus": "Display: 'c² + 2ab = a² + b² + 2ab'. Then, animate the cancellation of '2ab' from both sides, leaving: 'c² = a² + b²'. This final equation is highlighted prominently."
}
],
"visualSpec": {
"elements": [
"Right-angled triangle (vertices A, B, C; right angle at C)",
"Sides labeled: 'a' (opposite A), 'b' (opposite B), 'c' (opposite C - hypotenuse)",
"Squares built on sides a, b, c (areas a², b², c²)",
"Large square (side a+b)",
"Four identical right triangles (copies of the original)",
"Central square (side c) formed by the first arrangement of triangles",
"Two smaller squares (sides a, b) formed by the second arrangement of triangles",
"Two rectangles (dimensions a x b) formed by the second arrangement of triangles",
"Text labels for areas and equations"
],
"interactions": [
"Slider to control visibility/size of squares in Step 2",
"Next button to advance through steps and trigger animations/text reveals",
"Highlighting of specific visual elements (e.g., sides, squares, triangles) as they are discussed"
],
"mathLogic": "Area of a square = side * side; Area of a triangle = 1/2 * base * height; (a+b)² = a² + 2ab + b²; Equating areas: c² + 2ab = a² + b² + 2ab => c² = a² + b²"
}
},
"sourceCode": "\n\n\n\n \n \n Pythagorean Theorem Visual Proof\n \n\n\n\n\n