Answer
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Hint:Here in this question, we have to find the perimeter of the logo. The given logo is made from a rectangle and a major sector of the circle. For this, we need to find the perimeter of rectangle by adding a 2 length and 1 breadth and length of arc by using a formula \[Arc\,length = \dfrac{\theta }{{360}} \times 2\pi r\] and further to find the perimeter of logo add the perimeter of rectangle, length of arc and length of the line \[OD\] to get the required solution.
Complete step by step answer:
In geometry, perimeter can be defined as the path or the boundary that surrounds a shape. It can also be defined as the length of the outline of a shape. Consider the question: Given, the diagram of a company logo made from a rectangle of a length \[AB = OC = 1.5\,\,cm\] and breadth \[BC = OA = 0.5\,\,cm\] and a major sector of circle of radius \[OA = 0.5\,\,cm\].
We have to find the perimeter of the logo?
Perimeter of logo \[ = \] Perimeter of rectangle \[ + \] Arc length \[ + \] length of line \[OD\].
The perimeter of rectangle in logo \[ = OC + CB + AB\]
From figure, \[AB = OC = 1.5\,\,cm\] and \[BC = OA = 0.5\,\,cm\], then
\[ \Rightarrow \text{Perimeter of rectangle in logo}= 1.5 + 1.5 + 0.5\]
\[\therefore \text{Perimeter of rectangle in logo}=3.5\,\,cm\]
Therefore, the perimeter of the rectangle is \[3.5\,\,cm\]. Now find the arc length by using a formula:
\[Arc\,length = \dfrac{\theta }{{360}} \times 2\pi r\]
Where, \[\theta \] is the central angle of the arc and \[r\] is the radius of the circle.
Find the value of central angle \[\theta \]
In \[\vartriangle \,OED\]
\[\cos \theta = \dfrac{{EO}}{{DO}}\], then
\[\Rightarrow \cos \left( {\angle EOD} \right) = \dfrac{{0.25}}{{0.5}} = 0.5\]
\[\Rightarrow \angle EOD = {60^ \circ }\]
The central angle is the angle subtended by an arc of a sector at the centre of a circle.
\[\angle EOD\] in sector \[AOD\]\[ = {360^ \circ } - \left( {{{90}^ \circ } + {{60}^ \circ }} \right)\]
\[\Rightarrow \angle EOD\] in sector \[AOD\]\[ = {360^ \circ } - {90^ \circ } - {60^ \circ }\]
\[\Rightarrow \angle EOD\] in sector \[AOD\]\[ = {210^ \circ }\]
\[\Rightarrow \theta = {210^ \circ }\] and \[r = 0.5\], then arc length is
\[ \Rightarrow \,\,Arc\,length = \dfrac{{210}}{{360}} \times 2 \times \pi \times 0.5\]
\[ \Rightarrow \,\,Arc\,length = \dfrac{{21}}{{36}} \times 3.14\]
On simplification, we get
\[Arc\,length = 1.83\,\,cm\]
Now, the perimeter of given logo is:
Perimeter of logo \[ = \] Perimeter of rectangle \[ + \] Arc length \[ + \] length of line \[OD\].
On substituting the values, we have
\[\text{Perimeter of logo} =3.5 + 1.83 + 0.5\]
\[\therefore \text{Perimeter of logo} =5.83\,\,\,cm\]
Therefore, the perimeter of the given company logo is \[5.83\,\,\,cm\].
Note:The perimeter of any shape is found by measuring the length of the outer boundary of the shape. If a given figure is complex, try to identify the possible shapes and find them perimeter separately to make the problem easy. While doing a measurement based problem, do not forget to write the unit with the final answer, the unit for the perimeter will be the same as the unit of the length.
Complete step by step answer:
In geometry, perimeter can be defined as the path or the boundary that surrounds a shape. It can also be defined as the length of the outline of a shape. Consider the question: Given, the diagram of a company logo made from a rectangle of a length \[AB = OC = 1.5\,\,cm\] and breadth \[BC = OA = 0.5\,\,cm\] and a major sector of circle of radius \[OA = 0.5\,\,cm\].
We have to find the perimeter of the logo?
Perimeter of logo \[ = \] Perimeter of rectangle \[ + \] Arc length \[ + \] length of line \[OD\].
The perimeter of rectangle in logo \[ = OC + CB + AB\]
From figure, \[AB = OC = 1.5\,\,cm\] and \[BC = OA = 0.5\,\,cm\], then
\[ \Rightarrow \text{Perimeter of rectangle in logo}= 1.5 + 1.5 + 0.5\]
\[\therefore \text{Perimeter of rectangle in logo}=3.5\,\,cm\]
Therefore, the perimeter of the rectangle is \[3.5\,\,cm\]. Now find the arc length by using a formula:
\[Arc\,length = \dfrac{\theta }{{360}} \times 2\pi r\]
Where, \[\theta \] is the central angle of the arc and \[r\] is the radius of the circle.
Find the value of central angle \[\theta \]
In \[\vartriangle \,OED\]
\[\cos \theta = \dfrac{{EO}}{{DO}}\], then
\[\Rightarrow \cos \left( {\angle EOD} \right) = \dfrac{{0.25}}{{0.5}} = 0.5\]
\[\Rightarrow \angle EOD = {60^ \circ }\]
The central angle is the angle subtended by an arc of a sector at the centre of a circle.
\[\angle EOD\] in sector \[AOD\]\[ = {360^ \circ } - \left( {{{90}^ \circ } + {{60}^ \circ }} \right)\]
\[\Rightarrow \angle EOD\] in sector \[AOD\]\[ = {360^ \circ } - {90^ \circ } - {60^ \circ }\]
\[\Rightarrow \angle EOD\] in sector \[AOD\]\[ = {210^ \circ }\]
\[\Rightarrow \theta = {210^ \circ }\] and \[r = 0.5\], then arc length is
\[ \Rightarrow \,\,Arc\,length = \dfrac{{210}}{{360}} \times 2 \times \pi \times 0.5\]
\[ \Rightarrow \,\,Arc\,length = \dfrac{{21}}{{36}} \times 3.14\]
On simplification, we get
\[Arc\,length = 1.83\,\,cm\]
Now, the perimeter of given logo is:
Perimeter of logo \[ = \] Perimeter of rectangle \[ + \] Arc length \[ + \] length of line \[OD\].
On substituting the values, we have
\[\text{Perimeter of logo} =3.5 + 1.83 + 0.5\]
\[\therefore \text{Perimeter of logo} =5.83\,\,\,cm\]
Therefore, the perimeter of the given company logo is \[5.83\,\,\,cm\].
Note:The perimeter of any shape is found by measuring the length of the outer boundary of the shape. If a given figure is complex, try to identify the possible shapes and find them perimeter separately to make the problem easy. While doing a measurement based problem, do not forget to write the unit with the final answer, the unit for the perimeter will be the same as the unit of the length.
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