Question

If the diagonal of a square shrinks to
50
%
, then to what percentage would the area of the square shrink?

12
25
50
75
​

Answer 1

Given,

The diagonals shrinks to 50%

To find,

Percentage shrink of the area of the square.

Solution,

Let, the length of the diagonal = x unit

[ Assume, x as a variable to do the further mathematical calculations.]

Initial length of one side of the square = x/✓2 unit

Initial area of the square = (x/✓2)² = x²/2 sq.unit

Now,

Final length of the diagonal = x - (x×50/100) = x-x/2 = x/2 unit

Final length of the side of the given square = x/2 ÷ ✓2 = x/2✓2 unit

Final area = (x/2✓2)² = x²/8 sq.unit

Percentage decrease = 100 × (x²/8 ÷ x²/2) = 100 × (x²/8 × 2/x²) = 100/4 = 25%

Hence,25% of the area will be decreased.

Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

$\sf{Length \: of \: the \: side \: of \: a \: square }$

$= \displaystyle \: \sf{\frac{Length \: of \: the \: diagonal \: }{ \sqrt{2} } }$

GIVEN

If the diagonal of a square shrinks to 50 %

TO DETERMINE

The percentage in which the area of the square shrink

CALCULATION

Let x be the side of the square

Then the area of the square is

$\sf{ \: = {x}^{2} \: }$

Now the length of the diagonal

$= \displaystyle \: \sf{x \sqrt{2} }$

If the diagonal of a square shrinks to 50 %

Then the new length of the diagonal after shrinking

$= \displaystyle \: \sf{ x \sqrt{2 } - \bigg(x \sqrt{2} \times \frac{50}{100} \bigg) }$

$= \displaystyle \: \sf{ \frac{x}{ \sqrt{2} } }$

So the new length of the side after shrinking

$= \displaystyle \: \sf{ \frac{x}{2 } }$

So the new area of the square is

$= \displaystyle \: \sf{ { \bigg( \frac{x}{2} \: \bigg)}^{2} \: }$

$= \displaystyle \: \sf{ \frac{ {x}^{2} }{4} }$

So the shrinking in the area of the square

$= \displaystyle \: \sf{ {x}^{2} - \frac{ {x}^{2} }{4} }$

$= \displaystyle \: \sf{ \frac{ 3{x}^{2} }{4} }$

So the percentage in shrinking of the area is

$= \displaystyle \: \sf{ \frac{\frac{3 {x}^{2} }{4}}{ {x}^{2} } \times 100 \%}$

$= \displaystyle \: \sf{ 75 \%}$

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