1.SQUARE
2.PYTHAGOREAN
3.PERFECT SQUARE
4.TRIPLET
5.SQUARE ROOT
6.FACTOR

According to Pythagoras theorem,
(Hypotenuse)² = (Base)² + (Perpendicular)²
Clearly, the side opposite to 90° is hypotenuse
AC = Hypotenuse
AB = Base
BC = Perpendicular

(AC)²  = (AB)² + (BC)²
\(\Rightarrow\) (AC)² =(12)² +(5)²
\(\Rightarrow\) (AC)²=144+25
\(\Rightarrow\) (AC)² =169
\(\Rightarrow\) AC =\(\sqrt { 169 } \) = 13 cm

Let length of the square plot be a, then the area of square = a²
According to the question,
\(Area=101\frac { 1 }{ 400 } { m }^{ 2 }\)
\(\Rightarrow \ { a }^{ 2 }=101\frac { 1 }{ 400 } \)
\(\Rightarrow \ { a }^{ 2 }=\frac { 40401 }{ 400 } \)
\(\therefore \ a=\frac { 201 }{ 20 } \)
\(=10\frac { 1 }{ 20 } m\)
Hence, length of one side of the plot is 10\(\frac { 1 }{ 20 } \) m.

Let side of two squares be a₁ and a₂·
Then, perimeter of first square = 4a₁
According to the question,
 Perimeter = 40
\(\therefore\) 4a₁ = 40 \(\Rightarrow\) a₁ = 10
Similarly, second square's perimeter = 4a₂
But according to the question,
Perimeter = 96
\(\therefore\) 4a₂ = 96 \(\Rightarrow\) a₂ = 24
Let a be the side of another square.
Now, sum of the areas of first and second square area of
= Area of (first square) + (Area of second square)
=\({ a }_{ 1 }^{ 2 }+{ a }_{ 2 }^{ 2 }\)
Area =(10)² + (24)² =100 + 576
\(\therefore\) Area = 676
\(\Rightarrow\)a² = 676 \(\Rightarrow\) a = \(\sqrt { 676 } \)
Then, a=26 m
Another square's perimeter = 4a = 4 x 26 = 104 m
So, perimeter of another square is 104 m.

Here, 324 -1 = 323,323 - 3 = 320
320 - 5 = 315,315 -7 = 308
308 - 9 = 299, 299 - 11 = 288
288 -13 = 275,275 -15 = 260
260 - 17 = 243, 243 - 19 = 224
224 - 21 = 203,203 - 23 =180
180 - 25 =155, 155 - 27 =128
128 - 29 = 99, 99 - 31 = 68
68 - 33 = 35, 35 - 35 = 0
So, to get 0, we use 18 steps.
Hence, square root of 324 is 18.

We have, 1500

We see that, 38² < 1500 < 39²
So, number to be added = 39² - 1500
= 1521 - 1500 = 21
Therefore, the perfect square is 1500 + 21 = 1521 and \(\sqrt { 1521 } \)= 39
So, the required number is 21 and the square root is 39.

Let the breadth of the field be \(\times\) m. Then, length of the field will be 2 \(\times\) m.
Area of the rectangular field = Length \(\times\) Breadth
\(\therefore \ =(2x)\times (x)=({ 2x }^{ 2 }){ m }^{ 2 }\)
Given that, area = 2450 m²
Therefore, 2x²=2450
\(\Rightarrow \ { x }^{ 2 }=\frac { 2450 }{ 2 } \)
\(\Rightarrow \ x=\sqrt { 1225 } \Rightarrow \ x=35\ m\)
\(\therefore \) Breadth = 35 m and length = 35 \(\times\)2 = 70 m
Perimeter of the field = 2(l + b) = 2(70 + 35) m
= 2 \(\times\)105 m = 210 m

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