Question

The diagram shows a company logo made from a rectangle and a major sector of the circle: the circle has centre \[O\] and radius \[OA\]. \[OA = OD = 0.5\,\,cm\] and \[AB = 1.5\,\,cm\]. \[E\] is a point on \[OC\] such that \[OE = 0.25\,\,cm\] and angle \[OED = {90^ \circ }\]. Calculate the perimeter of logo

Answer 1

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.