Pythagorean Theorem

History of Pythagorean Theorem Review The Pythagorean theorem takes its name from the ancient Greek mathematician Pythagoras (569 B.C.?-500 B.C.?), who was perhaps the first to offer a proof of the theorem. But people had noticed the special relationship between the sides of a right triangle long before Pythagoras. The Pythagorean theorem states that the sum of the squares of the lengths of the two other sides of any right triangle will equal the square of the length of the hypotenuse, or, in mathematical terms, for the triangle shown at right, a 2 + b 2 = c 2. Integers that satisfy the conditions a 2 + b 2 = c 2 are called "Pythagorean triples."

History of Pythagorean Theorem Review We do not know for sure how Pythagoras himself proved the theorem that bears his name because he refused to allow his teachings to be recorded in writing. But probably, like most ancient proofs of the Pythagorean theorem, it was geometrical in nature. That is, such proofs are demonstrations that the combined areas of squares with sides of length a and b will equal the area of a square with sides of length c, where a, b, and c represent the lengths of the two sides and hypotenuse of a right triangle. Pythagoras himself was not simply a mathematician. He was an important philosopher who believed that the world was ruled by harmony and that numerical relationships could best express this harmony. He was the first, for example, to represent musical harmonies as simple ratios.

Bhaskara’s Proof: The Bhaskara figure contains a small square within a larger square and forms four right triangles inside the larger square. The length of the sides of the larger square are c, and the lengths of the legs of the right triangles are a and b. An easy deduction leads to the smaller square's sides being b - a. Thus, the area of the larger square can be used to prove the Pythagorean Theorem:

Assume the large object with sides c is a square with area c^2. Assume the center object is a square with area (b – a)^2. The area of a triangle is ½(base)(height) therefore the four triangles area is 4(1/2)(ab). The base is b and the height is a. Question: Could the base be a and the height b? Does it matter? Since the area of the triangle c contains the 4 triangles and the smaller square then:

Here is a very nice animation of a similar proof which can help you visualize how Bhaskara's proof works:

Using Similar Triangles to Derive The Pythagorean Theorem Assume BC=a, AB=c, and AC=b. If ABC is a right triangle with legs of length a & b and hypotenuse of length c then c^2= a^2 + b^2. There are 3 triangles in the diagram. ABC, ACH, & BCH. We need to show that these 3 triangles are similar.

Using Similar Triangles to Derive The Pythagorean Theorem If triangle ABC has a right angle C and triangle CHA has a right angle H both triangles share angle A. Therefore triangle ABC and triangle CHA share 2 common angles by AA criterion the angle C and the angle B must be equal. By the same argument triangle ABC and triangle BCH have congruent angles and have similar angles.

Using Similar Triangles to Derive The Pythagorean Theorem Lets set up some ratios: We already know that BC=a, AB=c, and AC=b. Let AH=d and HB=e. Try to find ratios such that when they are multiplied together and simplified we get a^2 and b^2. By definition of similar triangles: b/c= d/b which becomes b^2=cd. a/c=e/a which becomes a^2= ce. So if we add them together we get: a^2 + b^2 = cd + ce. Simplify: a^2 + b^2 = c(d + e) Notice that c = d + e then we get a^2 + b^2 = c^2 as desired.

GeoGebra Student Exploration Instructions 1.) Open GeoGebra and select Geometry from the Perspective panel. 2.) Select the Segment between Two Points tool and click two distinct places on the Graphics view to construct segment AB. 3.) If the labels of the points are not displayed, click the Move tool, right click each point and click Show label from the context menu. 4.) Next, we construct a line perpendicular to segment AB and passing through point B. To do this, choose the Perpendicular line tool, click on segment AB, then click on point B. 5.) Next, we create point C on the line. To do this, click the New point tool and click on the line. Be sure that the label of the third point is displayed.

GeoGebra Student Exploration – Point C on the line passing through B You have to be sure that C is on the line passing through B. Be sure that you cannot drag point C out of the line. Otherwise, delete the point and create a new point C. 6.) Hide everything except the three points by right clicking them and unchecking the Show Object option. 7.) Next, we rename point B to point C and vice versa. To rename point B to C, right click point B, click Rename and then type the new name, in this case point C, in the Rename text box, then click the OK button. Now, rename B (or B1) to C. 8.) Next we construct a square with side AC. Click the Regular polygon tool, then click on point C and click on point A. 9.) In the Points text box of the Regular polygon tool, type 4. If the position of the square is displayed the wrong way (right hand side of AC) just undo button and reverse the order of the clicks when creating the polygon. Figure 3 – Square containing side AC

GeoGebra Student Exploration 10. ) With the Polygon tool still active, click point B and click point C to create a square with side BC. Similarly, click point A, then click point B to create a square with side AB. After step 10, your drawing should look like the one shown below. Squares containing sides of right triangle ABC 11.) Hide the label of the sides of the side of the squares. 12.) Rename the sides of the rectangle as shown below. Triangle ABC with side lengths a, b and c.

GeoGebra Student Exploration 13.) Now, let us reveal the area of the three squares. Right click the interior of the square with side AC, then click Object Properties from the context menu to display the Preferences window. 14.) In the Basic tab of the Preferences window, check the Show Label check box and choose Value from the drop-down list box. Do this to the other two squares as well preferences-window Properties of squares shown in the Preferences window.

GeoGebra Student Exploration 15.) Move the vertices of the triangle. What do you observe about the area of the squares? 16.) You may have observed that the area of the biggest square is equal to the sum of the areas of the two smaller squares. To verify this, we can put a label in the GeoGebra window displaying the areas of the three squares. 17.) Suppose the side of the two smaller squares are a and b, and the side of the biggest square is c, what equation can you make to express the relationship of the of the three squares? 18.) What conjecture can you make based on your observation?

Pythagorean Theorem As we have seen from Bhaskara’s proof and the Similar Triangle Proof the Pythagorean Theorem is c^2 = a^2 + b^2. Let’s try some problems to see how this works: Pass out Pythagorean Triangle Examples worksheet. Work through the problems and feel free to ask any questions. This is your time to explore the theorem and put your knowledge into practice.

Example Problem #1 a = 8 & c =10 Given c^2=a^2+b^2 Pythagorean Theorem 10^2=8^2+b^2 Insert a & c into formula 100=64+b^2 Square numbers =64-64+b^2 Subtract 64 from both sides 36=b^2 Results of subtraction/Take square root of 36 b=6 Answer

Example Problem #2 b = 39 & c =89 Given c^2=a^2+b^2 Pythagorean Theorem 89^2=39^2+39^2 Insert b & c into formula 7921=a^2+1521^2 Square numbers = a^ Subtract 1521 from both sides 6400= a^2 Results of subtraction/Take square root of 6400 a = 80 Answer

Pythagorean Theorem Homework Please complete homework sheets Pythagorean Problems 1, 2 & 3 Notice there are 2 extra credit problems. Calculators are allowed. This is an individual assignment. This assignment is due at the beginning of the next class period. Check out these websites for more information and examples of the Pythagorean Theorem rean-theorem-game.php play.com/Pythagorean-Theorem- Jeopardy/Pythagorean-Theorem- Jeopardy.html theorem-game.html