Spatial Geometry & Functions · Common Core Grade 8 Geometry (Pythagorean Theorem), GCSE Maths (Right-Angled Triangles), Euclid's Elements Book I Proposition 47.
Pythagorean Theorem Visual Proof & Chord Diagram Solver Simulator
The iconic a² + b² = c² has over 400 distinct proofs. Drag legs a and b to dynamically inspect Zhao Shuang's four congruent right triangles expanding into the hypotenuse square, or animate Euclid's sheared shears.
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Parameter Controls
cm
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method
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Live Derivations & Metrics
hypotenuse_c
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c = √(a² + b²)
area_c2
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Area of square on hypotenuse
tri_area_sum
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4 × (1/2 ab) = 2ab
inner_square_area
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(b - a)²
Real-Time Formula Derivation
Governing Formula:
--Substitution:
--Regime: --
Worked Examples & Exam Question Deep Dive
Zhao Shuang Chord: Outer Area 25, Inner Area 1, Find Leg Product ab
Problem: In a Zhao Shuang chord diagram formed by 4 congruent right triangles, the outer large square area is 25 and the central inner square area is 1. Find product a·b and sum (a+b).
Analytical Strategy: Outer area c² = 25, so c = 5. Inner square (b-a)² = 1. Total area of 4 triangles = 25 - 1 = 24, giving 2ab = 24 => ab = 12. Using (a+b)² = c² + 2ab = 25 + 24 = 49, we get a+b = 7.
Step-by-Step Derivation:
- Step 1: In Zhao Shuang diagram: c² = 4 × (1/2 ab) + (b-a)².
- Step 2: Given c² = 25, (b-a)² = 1, so 2ab = 25 - 1 = 24.
- Step 3: Solve for product ab = 12.
- Step 4: Compute sum square: (a+b)² = a² + b² + 2ab = c² + 2ab = 25 + 24 = 49.
- Step 5: Take square root: a + b = 7 (the sides are 3 and 4).
⚠️ Common Misconception Pitfall: Confusing side lengths: outer square side is hypotenuse c, while inner square side is the difference of legs (b-a).
Student Inquiry Worksheet & Practice Problems
Q1. Right triangle with legs 5 and 12 has what hypotenuse length?
Answer Key: 13 (√(5² + 12²) = √169 = 13)
Q2. In Garfield's proof, the trapezoid area 1/2(a+b)² equals the sum of 3 right triangles. What equation results?
Answer Key: 1/2 (a+b)² = 2 × (1/2 ab) + 1/2 c² => a² + b² = c²
Frequently Asked Questions & Misconceptions
Why was the Zhao Shuang chord diagram chosen as the logo for the 2002 International Congress of Mathematicians (ICM)?
Created by 3rd-century Chinese mathematician Zhao Shuang in commentary on the Zhoubi Suanjing, it represents a world-class masterpiece of visual geometric proof harmonizing algebra and geometry.