7日間コース
Master the Pythagorean Theorem in 7 Days
From basic calculations to special triangles, 3D diagonals, and shortest surface paths — master the most powerful theorem in geometry.
対象: Junior High 3 / Overseas Japanese students
Concepts & Worked Examples
Pythagorean theorem, special triangles (45-45-90, 30-60-90), distance formula, 3D applications, and surface shortest paths.
直角三角形で、斜辺の2乗 = 他の2辺の2乗の和
a² + b² = c²（cが斜辺）
例：2辺が3と4 → 斜辺 = √(9+16) = √25 = 5 例：斜辺13、1辺5 → もう1辺 = √(169−25) = √144 = 12
a² + b² = c² が成り立てば直角三角形。 例：5, 12, 13 → 25 + 144 = 169 = 13² ✓
辺の比：1 : 1 : √2 例：辺 = 5 → 斜辺 = 5√2
辺の比：1 : √3 : 2 例：最短辺(30°の対辺) = 4 → 中辺 = 4√3、斜辺 = 8
(x₁, y₁) から (x₂, y₂) → d = √((x₂−x₁)² + (y₂−y₁)²)
例：A(2, 1), B(−5, −2) → d = √(49+9) = √58
a × b × c → 対角線 = √(a² + b² + c²) 例：3×4×5 → √50 = 5√2
母線と底面の半分で直角三角形を作る。 例：底面の1辺6、母線5 → 高さ = √(25−9) = 4
ポイント： 展開図を描いて、直線で結ぶ。
例：直方体の表面をBからAまで最短で移動。 展開図上で直線距離 = √((3+4)² + 7²) = 7√2
いくつかの展開の仕方を試して、最も短いものを選ぶ。
A right triangle has legs 6 and 8. Find the hypotenuse.
解答
c = √(36 + 64) = √100 = 10
Classic 3-4-5 triple scaled by 2: 6-8-10.
In a 30-60-90 triangle, the hypotenuse is 14. Find all sides.
解答
Shortest (30°) = 14/2 = 7 Medium (60°) = 7√3 Hypotenuse = 14
Ratio 1:√3:2. Divide hypotenuse by 2 for shortest side.
Find the diagonal of a rectangular prism with dimensions 1 × 2 × 2.
解答
d = √(1 + 4 + 4) = √9 = 3
Space diagonal = √(a² + b² + c²).
Find the distance between points A(1, 3) and B(4, 7).
解答
d = √((4−1)² + (7−3)²) = √(9+16) = √25 = 5
Use the distance formula, which is the Pythagorean theorem on coordinates.
Basic Practice
Basic Pythagorean calculations and special triangles.
三平方の定理の基礎を練習。
Right triangle: legs 5 and 12. Find hypotenuse.
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答え
13
Right triangle: hypotenuse 10, one leg 6. Find the other.
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答え
√(100−36) = √64 = 8
Is a triangle with sides 7, 24, 25 a right triangle?
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答え
Yes: 49 + 576 = 625 = 25²
45-45-90 triangle with hypotenuse 10. Find legs.
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答え
10/√2 = 5√2 each
Distance from (0,0) to (3,4).
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答え
√(9+16) = 5
Standard Practice
Special triangles and coordinate distance. ★2-3.
Pattern Practice
All Pythagorean theorem patterns. ★2-3.
Applied Practice
3D diagonals, heights, and area problems. ★3-4.
Advanced Practice
Surface shortest paths and complex 3D applications. ★4-5.
Final Test & Bridge to Next
Comprehensive test on all Pythagorean theorem applications.