Pythagoras and Basic Proportionality

Also called

Pythagoras theorem · Pythagorean theorem · a² + b² = c² · 3-4-5 triangle · hypotenuse · square on the hypotenuse · Perigal dissection · Basic Proportionality Theorem · Thales theorem · BPT · line parallel to one side of a triangle · similar triangles · similarity of triangles · AAA similarity · corresponding sides in the same ratio · converse of Pythagoras · पाइथागोरस प्रमेय · आधारभूत आनुपातिकता प्रमेय · समरूप त्रिभुज · कर्ण

Topics

• In a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides: a² + b² = c².
• Legs 3 and 4 give hypotenuse 5, because 9 + 16 = 25: the tiles of the two small squares exactly fill the big one.
• Perigal’s cut: four equal pieces of the larger leg’s square plus the smaller square fill the hypotenuse square.
• Water that fills the two leg tanks fills the hypotenuse tank exactly: the areas add up.
• Basic Proportionality Theorem: a line parallel to one side of a triangle divides the other two sides in the same ratio.
• If DE ∥ BC in triangle ABC, then AD/DB = AE/EC; the converse also holds.
• Two triangles are similar when their corresponding angles are equal and their corresponding sides are in the same ratio.
• Triangle ADE cut off by a parallel line is similar to triangle ABC, so similar triangles nest at a shared angle.
• Converse: if a triangle’s sides satisfy a² + b² = c², the angle opposite the longest side is a right angle.

About this model

A GLB file of about 358 KB, shown in Lably's 3D viewer and, on a phone, placed in your room with AR. Credit: Model, science and animation: Lably (built procedurally with three.js; no downloaded meshes). | CC BY 4.0. Original geometry and animation made for Lably.

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