Proof Without Words - Part 4
The image above illustrates which identity?
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15 solutions
The whole figure is divided into 4 parts. ( 2 squares and 2 rectangles)
Area of square:
With side a = a 2
With side b = b 2
There are two rectangles of same measurements. L e n g t h = a , B r e a d t h = b
Thus, area of 2 rectangles = 2 × ( l × b ) = 2 a b
So, total area = a 2 + b 2 + 2 a b
Thus, the answer to this question is the identity:
( a + b ) 2 = a 2 + 2 a b + b 2
Or else we can directly put the identity:
Side of square = a + b
Area of square = ( a + b ) 2 = a 2 + 2 a b + b 2
Good pictorial depiction of a very basic formula. Great!!! The only regret is that had i got such things in my school days, I would have loved mathematics much more ! ;-)
very well explained~ thanks :D
Yeah, that's the same idea I used... But did anyone use some other way to figure this out? Just curious... Really nice problem and solution, by the way. Thumbs up!
yep the same thing i got it !
excellent !!!
False + Mean less problime.
The trick is just analyze that the image involves a quadratic sum: a^2 + ab + ab + b^2 = a^2 + 2ab + b^2 = (a+b)^2
whole figure is about area of square which have its side as (a+b)= area of square having side "a" + area of square having side "b" + area of two rectangles having respective area as "ab"...so the equation as follows
area of square (a+b)= (a+b)^2
area of square(a)= a^2
area of square (b)= b^2
area of two same rectangles = 2x(ab)
as per quatation
(a+b)^2= a^2 +b^2 +2ab
Excellent
The area of the small red square is b^2; the area of the rectangles is ‘ab’ and we have two of them hence total area is ‘2ab’; the area of big blue square is a^2. The sum of all this represent the area of the big square formed by these small shapes. The area of the big square is (a+b)^2.. so equate the sum of small parts to the whole
The whole figure is a square with side length (a+b). The top left and bottom right rectangles both have length a and height b giving them an area of ab each and a total area of 2ab The bottom left square has side length a so its area is a a The top right square has side length b making its area b b
The sum of all of these areas is equal to the area of the larger square with side length (a+b) which is (a+b)(a+b). Therefore, the answer is (a+b)(a+b)=a a+2ab+b b
Area sq = L^2 & L=a+b
so A=L^2=(a+b)^2
(a+b)^2= a^2 +b^2+2ab :)
the surface of square is a^2....since a=a+b then the surface is (a+b)^2
The figure is a square of length a+b. Then total area=(a+b)^2=area of square of side a+2(area of rectangles of sides a and b)+ area of square of side b.
Como na imagem está expicitado que cada lado tem sua medida, fica fácil saber que o quadrado de maior lado é a² e o de menor lado é b², resta os retangulos que são iguais somando juntos 2ab, o que nos dá: a²+b²+2ab=(a+b)²
there are four areas are in figure = a^2 , b^2 , ab , ab
so total = (a+b)^ 2
Sum of: areas of 2 pink rectangles (ab +ab) + Area of red square (b^2) + Area of grey sq. (a^2) =Area of the biggest sq. (a+b)^2 Thus we get the required relation.
Its a square one part is a another is b, So none hand is a+b and a+b, So area is (a+b)(a+b)= (a+b)2 = a2 +2ab+b2
Area of Square= s i d e 2 ; Here we have four sides(one side = a+b) which are equal hence reprsents square as ( a + b ) 2 = a 2 + 2 a b + b 2
the pink is b and a is blue so it is clearly divided wich is a2+2ab=b
A and b together makes a side of square so we get l.h.s (a+b)^2 then area of 2nd large square whose side is a is a^2, add of area of rectangle having sides a and b is 2ab and smallest square having side b is b^2 so this makes whole eq
The whole figure is a square with side length ( a + b ) . The top left and bottom right rectangles both have length a and height b , giving them an area of a b each and a total area of 2 a b . The bottom left square has side length a , so its area is a 2 . The top right square has side length b , making its area b 2 .
The sum of all of these areas is equal to the area of the larger square with side length ( a + b ) , which is ( a + b ) 2 . Therefore, ( a + b ) 2 = a 2 + 2 a b + b 2