Transcript Bhaskara’s Proof The Bhaskara figure contains a small

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, a2 + b2 =
c2. Integers that satisfy the conditions a2 + b2 = c2 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
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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
100-64=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
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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
7921-1521= a^2+1521-1521 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
• http://www.kidsnumbers.com/pythago
rean-theorem-game.php
• http://www.mathplay.com/Pythagorean-TheoremJeopardy/Pythagorean-TheoremJeopardy.html
• http://crctlessons.com/Pythagoreantheorem-game.html