This article was reviewed by Joseph Meyer. Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University.
This article has been viewed 147,892 times.
Garfield was the 20th President in 1881 and did this proof of the Pythagorean Theorem while he was still a seated member of Congress in 1876. It is interesting to note that he was fascinated by geometry, like President Lincoln, but was not a professional mathematician or geometer.
Steps
The Tutorial
Construct a right triangle resting on side b with right angle to the left connected to upright and perpendicular side a, with side c connecting the endpoints of a and b.,br>
Construct a similar triangle with side b now extending in a straight line from the original side a, then with side a parallel along the top to the bottom original side b, and side c connecting the endpoints of the new a and b.
Understand the goal. We are interested to know the angle x formed where the two side c's meet. Thinking about it, the original triangle was made of 180 degrees with the angle on the right at the far end of b, called theta, and the other angle at the top of a, being 90 degrees minus theta, as all the angles total 180 degrees and we already have one 90 degree angle.
Transfer your angle knowledge to the upper new triangle. At the bottom, we have theta, at the top left we have 90 degrees, and the top right we have 90 degrees minus theta.- The mystery angle x is 180 degrees. So theta + 90 degrees-theta + x = 180 degrees. Adding theta and negative theta gives us zero on the left, and subtracting 90 degrees from both sides leaves x equal to 90 degrees. So we have established that the mystery angle x = 90 degrees.
- The mystery angle x is 180 degrees. So theta + 90 degrees-theta + x = 180 degrees. Adding theta and negative theta gives us zero on the left, and subtracting 90 degrees from both sides leaves x equal to 90 degrees. So we have established that the mystery angle x = 90 degrees.
Look at the whole figure as a trapezoid in two ways. First, the formula for a trapezoid is A= the Height x (Base1 + Base 2)/2. The height is a+b and (Base1 + Base 2)/2 = 1/2(a + b). So that all equals 1/2 (a+b)^2.
Look at the interior of the trapezoid and add up the areas, in order to set them equal to the formula just found. We have the two smaller triangles at bottom and left, and those together equal 2*1/2(a*b), which just equals (a*b). Then we also have 1/2 c*c, or 1/2 c^2. So together we have the other formula for the area of the trapezoid equaling (a*b)+ 1/2 c^2.
Set the two Area formulas equal. 1/2(a+b)^2=(a*b)+1/2 c^2. Now multiply both sides by 2 to get rid of the 1/2's 2(1/2 (a+b)^2) = 2((a*b)+ 1/2 c^2.) which simplifies as (a+b)^2 = 2ab + c^2.EXPERT TIPJoseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University.Math Teacher
Joseph Meyer
Math TeacherLearning different proofs will help you master the Pythagorean Theorem. Discovering the Pythagorean Theorem can be approached through visual or algebraic methods. By exploring the proof from different angles, you can solidify your knowledge and make it easier to remember.
Advertisement
Explanatory Charts, Diagrams, Photos
Expert Q&A
Video
Tips
- There are over 100 proofs of the Pythagorean Theorem -- maybe you can find a new one!Thanks
References
- http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.01.0086%3Abook%3D1%3Atype%3DProp%3Anumber%3D47 - resource source, content is shared on a Creative Commons Sharealike 3.0 license, Please refer esp. to Book I, Proposition 47
- Video: "James Garfield's proof of the Pythagorean Theorem." -khanacademy Published on Nov 27, 2012- LICENSE: Creative Commons (Attribution-Noncommercial-No Derivative Works).
About this article
Reader Success Stories
"I was so confused when my (very bad) math teacher tried to explain this by cramming it for the test the next day. I came here and through the actual explanations and diagrams I learned how to do it in time."..." more
Did this article help you?
⚠️ Disclaimer:
Content from Wiki How English language website. Text is available under the Creative Commons Attribution-Share Alike License; additional terms may apply.
Wiki How does not encourage the violation of any laws, and cannot be responsible for any violations of such laws, should you link to this domain, or use, reproduce, or republish the information contained herein.
- - A few of these subjects are frequently censored by educational, governmental, corporate, parental and other filtering schemes.
- - Some articles may contain names, images, artworks or descriptions of events that some cultures restrict access to
- - Please note: Wiki How does not give you opinion about the law, or advice about medical. If you need specific advice (for example, medical, legal, financial or risk management), please seek a professional who is licensed or knowledgeable in that area.
- - Readers should not judge the importance of topics based on their coverage on Wiki How, nor think a topic is important just because it is the subject of a Wiki article.
