Answer
Verified
478.2k+ views
Hint: Find the area of the given and find the area of the circle which is \[{{\dfrac{1}{5}}^{th}}\]of the total area. Take O as the central point. Find the value of OA, AE and find the value of arc EF. Add all the values to get the perimeter of part ABEF.
Complete step-by-step Solution:
Given a square ABCD, we know that in a square all sides are equal and all the angles are \[{{90}^{\circ }}\].
Given that AB = 22cm, where AB is the side of a square.
We are told that all sides are equal.
So, AB = BC = CD = DA = 22cm.
Area of the square is given as the square of the side.
i.e. Area of square \[={{\left( side \right)}^{2}}\].
Here, the sides of the square are AB, BC, CD and DA which is equal to 22cm.
\[\therefore \]Side of square = 22cm
\[\therefore \]Area of square \[={{\left( 22 \right)}^{2}}=22\times 22=484c{{m}^{2}}\]
It is said that the square ABCD is divided into 5 equal parts in terms of area.
We know the total area of the square as \[484c{{m}^{2}}\]. Out of these 5 equal parts, one part is the circle inscribed inside the square. To find the area of the circle we need to divide the total area by 5.
\[\therefore \]Area of circle\[=\dfrac{1}{5}\times \]area of square
\[=\dfrac{1}{5}\times 484=96.8c{{m}^{2}}\]
Area of the circle is given by the formula, \[\pi {{r}^{2}}\]. We need to find the radius, r of the circle.
\[\begin{align}
& \therefore \pi {{r}^{2}}=96.8 \\
& \Rightarrow {{r}^{2}}=\dfrac{96.8}{\pi }=30.81 \\
& \therefore r=\sqrt{30.81}=5.55cm \\
\end{align}\]
\[\therefore \]Radius of the central part = 5.55cm.
Let us consider O as the center of the circle thus O also becomes the center of the square.
We know the diagonal of a square is given by \[\sqrt{2}a\], where ‘a’ is the side of the square.
Let the diagonal of square AC = \[\sqrt{2}a\].
\[\begin{align}
& AC=OA+OC \\
& \therefore OA=OC=\dfrac{AC}{2} \\
\end{align}\]
i.e. Diagonals of the square are equal in length and bisect each other.
\[\therefore OA=\dfrac{\sqrt{2}a}{2}=\dfrac{\sqrt{2}\times 22}{2}=11\sqrt{2}cm\]
From the figure we can say that length of AE is equal to length of BF.
\[AE=OA-OE\][from the figure]
\[\therefore AE=BF=OA-OE\]
OE is equal to the radius of the circle = 5.55cm.
\[\begin{align}
& \therefore AE=BF=11\sqrt{2}-5.55\left[ \because \sqrt{2}=1.414 \right] \\
& AE=\left( 11\times 1.414 \right)-5.55 \\
& AE=BF=9.96cm \\
\end{align}\]
The arc of circle EF\[=\dfrac{1}{4}\times \]circumference of the circle
We know the circumference of the circle \[=2\pi r\].
\[\therefore arcEF=\dfrac{1}{4}\times 2\pi r=\dfrac{1}{4}\times 2\pi \times 5.55=8.72cm\]
Hence, perimeter of point ABEF = AB + AE + EF + BF = 22 + 9.96 + 8.72 + 9.96 = 50.64cm
\[\therefore \]Perimeter of part ABEF = 50.64cm.
Note:
Since it is a square, the area of part ABEF is equal to the area of part BFGC, CGHD and DHEF. And their areas are the same as mentioned in the question.
The area of the circle is\[96.8c{{m}^{2}}\], which was one part of the circle. Similarly, the area of ABEF, BFGC, CGHD and DHEF is equal to \[96.8c{{m}^{2}}\]. By adding all this we get the total area as \[484c{{m}^{2}}\].
Complete step-by-step Solution:
Given a square ABCD, we know that in a square all sides are equal and all the angles are \[{{90}^{\circ }}\].
Given that AB = 22cm, where AB is the side of a square.
We are told that all sides are equal.
So, AB = BC = CD = DA = 22cm.
Area of the square is given as the square of the side.
i.e. Area of square \[={{\left( side \right)}^{2}}\].
Here, the sides of the square are AB, BC, CD and DA which is equal to 22cm.
\[\therefore \]Side of square = 22cm
\[\therefore \]Area of square \[={{\left( 22 \right)}^{2}}=22\times 22=484c{{m}^{2}}\]
It is said that the square ABCD is divided into 5 equal parts in terms of area.
We know the total area of the square as \[484c{{m}^{2}}\]. Out of these 5 equal parts, one part is the circle inscribed inside the square. To find the area of the circle we need to divide the total area by 5.
\[\therefore \]Area of circle\[=\dfrac{1}{5}\times \]area of square
\[=\dfrac{1}{5}\times 484=96.8c{{m}^{2}}\]
Area of the circle is given by the formula, \[\pi {{r}^{2}}\]. We need to find the radius, r of the circle.
\[\begin{align}
& \therefore \pi {{r}^{2}}=96.8 \\
& \Rightarrow {{r}^{2}}=\dfrac{96.8}{\pi }=30.81 \\
& \therefore r=\sqrt{30.81}=5.55cm \\
\end{align}\]
\[\therefore \]Radius of the central part = 5.55cm.
Let us consider O as the center of the circle thus O also becomes the center of the square.
We know the diagonal of a square is given by \[\sqrt{2}a\], where ‘a’ is the side of the square.
Let the diagonal of square AC = \[\sqrt{2}a\].
\[\begin{align}
& AC=OA+OC \\
& \therefore OA=OC=\dfrac{AC}{2} \\
\end{align}\]
i.e. Diagonals of the square are equal in length and bisect each other.
\[\therefore OA=\dfrac{\sqrt{2}a}{2}=\dfrac{\sqrt{2}\times 22}{2}=11\sqrt{2}cm\]
From the figure we can say that length of AE is equal to length of BF.
\[AE=OA-OE\][from the figure]
\[\therefore AE=BF=OA-OE\]
OE is equal to the radius of the circle = 5.55cm.
\[\begin{align}
& \therefore AE=BF=11\sqrt{2}-5.55\left[ \because \sqrt{2}=1.414 \right] \\
& AE=\left( 11\times 1.414 \right)-5.55 \\
& AE=BF=9.96cm \\
\end{align}\]
The arc of circle EF\[=\dfrac{1}{4}\times \]circumference of the circle
We know the circumference of the circle \[=2\pi r\].
\[\therefore arcEF=\dfrac{1}{4}\times 2\pi r=\dfrac{1}{4}\times 2\pi \times 5.55=8.72cm\]
Hence, perimeter of point ABEF = AB + AE + EF + BF = 22 + 9.96 + 8.72 + 9.96 = 50.64cm
\[\therefore \]Perimeter of part ABEF = 50.64cm.
Note:
Since it is a square, the area of part ABEF is equal to the area of part BFGC, CGHD and DHEF. And their areas are the same as mentioned in the question.
The area of the circle is\[96.8c{{m}^{2}}\], which was one part of the circle. Similarly, the area of ABEF, BFGC, CGHD and DHEF is equal to \[96.8c{{m}^{2}}\]. By adding all this we get the total area as \[484c{{m}^{2}}\].
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE