File:Pythagoras similar triangles.svg

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Summary

Description

English: Similar triangles proof for Pythagoras' theorem.

In triangle ACB, angle ACB is the right angle. CH is a perpendicular on hypotenuse AB of triangle ACB.
In triangle AHC and triangle ACB, ∠AHC=∠ACB as each is a right angle. ∠HAC=∠CAB as they are common angles at vertex A. Thus triangle AHC is similar to triangle ACB by AA test.
Thus, ${\frac {AC}{AB}}={\frac {AH}{AC}}$ $\therefore {AC}^{2}={AB}\times {AH}$

In triangle BHC and triangle ACB, ∠BHC=∠ACB as each is a right angle. ∠HBC=∠CBA as they are common angles at vertex B. Thus triangle BHC is similar to triangle BCA by AA test.
Thus, ${\frac {BC}{AB}}={\frac {BH}{BC}}$ $\therefore {BC}^{2}={AB}\times {BH}$

$\therefore {AC}^{2}+{BC}^{2}=({AB}\times {AH})+({AB}\times {BH})$

$\therefore {AC}^{2}+{BC}^{2}={AB}\times ({AH}+{BH})$

$\therefore {AC}^{2}+{BC}^{2}={AB}\times {AB}$

\therefore {AC}^{2}+{BC}^{2}={AB}^{2}

which is the Pythagoras theorem.

Date

17 March 2012, 14:16 (UTC)

Source

This file was derived from: Pythagoras similar triangles.PNG:
Reference:

^[1]

↑ (2011) GEOMETRY Standard X, Secretary, Maharashtra State Board of Secondary and Higher Secondary Education, Pune-411 004, pp. 22, 23

Author

Pythagoras_similar_triangles.PNG: Brews ohare
derivative work: Gauravjuvekar

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English: A simplified version with lower-case notations(and related elements) removed

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Original upload log

This image is a derivative work of the following images:

File:Pythagoras_similar_triangles.PNG licensed with Cc-by-sa-3.0, GFDL
- 2010-05-05T04:59:48Z Brews ohare 960x600 (70261 Bytes) Arcs for angles
- 2010-05-04T23:02:02Z Brews ohare 960x600 (67020 Bytes) Simplify further
- 2010-05-04T22:35:26Z Brews ohare 960x600 (74056 Bytes) Move labels outside triangle
- 2010-05-04T22:26:16Z Brews ohare 948x598 (71817 Bytes) spacing of elements
- 2010-05-04T22:20:25Z Brews ohare 948x597 (71826 Bytes) Reduce clutter
- 2010-05-04T21:43:58Z Brews ohare 986x693 (98895 Bytes) larger angle labels; right angle; altitude heavier
- 2010-05-04T19:37:49Z Brews ohare 988x695 (98337 Bytes) larger labels
- 2010-05-04T19:26:26Z Brews ohare 986x693 (94166 Bytes) enlarge labels
- 2010-05-04T19:18:46Z Brews ohare 970x673 (90356 Bytes) {{Information |Description={{en|1=Similar triangles for Pythagoras' theorem}} |Source={{own}} |Author=[[User:Brews ohare|Brews ohare]] |Date=2012-05-04 |Permission= |other_versions= }} [[Category:Pythagorean theorem]]

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File history

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	Date/Time	Dimensions	User	Comment
current	12:17, 10 April 2012	811 × 660 (10 KB)	Gauravjuvekar	validation
	14:50, 17 March 2012	811 × 660 (15 KB)	Gauravjuvekar	Reverted to version as of 14:18, 17 March 2012, arrows
	14:32, 17 March 2012	512 × 417 (8 KB)	Gauravjuvekar	Removing line-height attribute (11 times)
	14:22, 17 March 2012	512 × 417 (8 KB)	Gauravjuvekar	validation
	14:18, 17 March 2012	811 × 660 (15 KB)	Gauravjuvekar	== {{int:filedesc}} == {{Information \|Description={{en\|1=Similar triangles for Pythagoras' theorem}} \|Source={{Derived from\|Pythagoras_similar_triangles.PNG\|display=50}} \|Date=2012-03-17 14:16 (UTC) \|Author=*File:Pythagoras_similar_triangles.PNG:...