Hi Can U Please Prove (a+b)^2
Answers
Answer:
- (a²+b²+2ab)
Step-by-step explanation:
1) The length of each side of first square is
a
. So, its area is equal to a²
2)The dimension of one rectangle are
b
and
a
. So, its area is equal to ba
3)The dimension of second rectangle are
a
and
b
. So, its area is equal to ab
4)The length of each side of second square is
b
. So, its area is equal to b²
Now, add areas of the all four geometrical shapes to express the whole area in mathematical form.
a square plus 2ab plus b square
a
2
+
b
a
+
a
b
+
b
2
According to the commutative property of multiplication, the product of
a
and
b
is equal to the product of
b
and
a
. The equality of the areas of both rectangles can also be proved geometrically.
Therefore, the term
b
a
can be written as
a
b
and vice-versa.
⟹
a
2
+
a
b
+
a
b
+
b
2
⟹
a
2
+
2
a
b
+
b
2
⟹
a
2
+
b
2
+
2
a
b
Equality of the Areas of shapes
a plus b whole square proof
We have derived that the area of a square is equal to
(
a
+
b
)
2
in the first step.
In the second step, it is proved that the sum of the areas of four geometric shapes is equal to
a
2
+
b
2
+
2
a
b
.
Actually, a square is divided as four geometrical shapes. It is obvious that the area of the square is equal to sum of the areas of them.
∴
(
a
+
b
)
2
=
a
2
+
b
2
+
2
a
b
Geometrically, it is proved that square of
a
+
b
can be expanded as
a
squared plus
b
squared plus two times product of
a
and
b
.