Question

There was a king. He had a square-size pond. He had planted trees at all the four corners of the pond. One day he asked the engineer that he wanted to double the area of his pond. But there were two conditions. First, the pond should remain a square and none of the trees at its four corners should get submerged in water. You tell how it is possible.

Answer 1

Given :

A square pond is made, whose each corner has a tree.

To find :

How to double the area od square pond, without submerging the trees.

Solution.

A square pond is made, let its each sides are equal to A.

Each corner has a grown tree.

Now king wants to double the area of pond with out submerging the tree.

So, he sould take all tress at the boundaries of new square which is to me made.

It means,

A new square will be made which will have corners of the first square in mid point of each side of new square.

So, that no tree will submerge in pond.

So,Triangles of equal shape will be made on each side of old square.

So Total area of these four triangle = Area of old square,

So, Area of new square will become, double of old square.

So,

Side of old square = A

Area of old square = A²

So, Area of 4 new triangles = A²

then area of one new triangle = 0.25 A²

We know that

Area of triangle :

$= \frac{1}{2} \times base \times height$

Putting

Value of area = 0.25 A²,

Base = A

in formula of area of triangle,

Height of triangle :

$0.25 \times A^2 = \frac{1}{2} \times A \times h \\ so \\ h = 0.5A \:$

so, Height of triangle is half of Side of old square,

We know that

Altitude of this triangle is equal to half of the Side of new square.

Because the corner of old square is the mid point of side of new square.

This is a isosceles triangle, and have 90° angle between half the distance of old square.

So,

By pythagorus theorem,

Altitude of triangle :

$= \sqrt{ (\frac{A}{2}) ^{2} + (\frac{A}{2}) ^{2} \: } = \frac{A}{\sqrt{2} } \:$

So,

Side of new Square = double of altitude of triangle = A√2.

So area of new square = (A√2)² = 2A².

Answer 2

Answer:

Make more triangles.

Step-by-step explanation:

Given :

A square pond is made, whose each corner has a tree.

To find :

How to double the area od square pond, without submerging the trees.

Solution.

A square pond is made, let its each sides are equal to A.

Each corner has a grown tree.

Now king wants to double the area of pond with out submerging the tree.

So, he sould take all tress at the boundaries of new square which is to me made.

It means,

A new square will be made which will have corners of the first square in mid point of each side of new square.

So, that no tree will submerge in pond.

So,Triangles of equal shape will be made on each side of old square.

So Total area of these four triangle = Area of old square,

So, Area of new square will become, double of old square.

So,

Side of old square = A

Area of old square = A²

So, Area of 4 new triangles = A²

then area of one new triangle = 0.25 A²

We know that

Area of triangle :

Putting

Value of area = 0.25 A²,

Base = A

in formula of area of triangle,

Height of triangle :

so, Height of triangle is half of Side of old square,

We know that

Altitude of this triangle is equal to half of the Side of new square.

Because the corner of old square is the mid point of side of new square.

This is a isosceles triangle, and have 90° angle between half the distance of old square.

So,

Side of new Square = double of altitude of triangle = A√2.

So area of new square = (A√2)² = 2A².