In this figure, if the sidelength of the square is doubled, what is the increase in the areas of the regions 1, 2 and 3? Give reasons. Class 8

In this figure, if the sidelength of the square is doubled, what is the increase in the areas of the regions 1, 2 and 3? Give reasons. Class 8

Question 1.

In this figure, if the sidelength of the square is doubled, what is the increase in the areas of the regions 1, 2 and 3? Give reasons. Class 8

Solution:

Let sidelengths of the square be a units.

So, BD² = a² + a² = 2a²

or BD = [latex]a \sqrt{2}[/latex] units.

So, AO = [latex]\frac{a \sqrt{2}}{2}[/latex]

If AD become 2a, then

BD² = 4a² + 4a² = 8a²

or BD = [latex]2 \sqrt{2}[/latex] a²

So, BD = [latex]\frac{2 \sqrt{2} a}{2}[/latex] = [latex]\sqrt{2} a[/latex]

Now, area of 1 = [latex]\frac{1}{2}[/latex] DO × AO

= [latex]\frac{1}{2}[/latex] × [latex]\frac{a \sqrt{2}}{2}[/latex] × [latex]\frac{a \sqrt{2}}{2}[/latex]

= [latex]\frac{1}{8}[/latex]2a² sq. units

= [latex]\frac{1}{4}[/latex]a² sq. units

and area of new region 1

= [latex]\frac{1}{2}[/latex]DO × AO

= [latex]\frac{1}{2}[/latex] × [latex]\sqrt{2} a[/latex] × [latex]\sqrt{2} a[/latex]

= [latex]\frac{1}{2}[/latex] × 2a² = a² sq. units

= 4 × [latex]\frac{1}{4}[/latex]a² sq. units

which is 4 times the previous region 1.

Similarly, area of each of the regions 2 and 3 also becomes 4 times of the previous area.