Question

In the diagram, AOB is a quarter circle with centre O and radius r. The point C lies on the arc AB  and the point D lies on OB. The line CD is parallel to AO and angle  radians.

     i.       Express the perimeter of the shaded region in terms of r,  and .

   ii.       For the case where r = 5 cm and  , find the area of the shaded region.

Solution

i.
 

It is evident from the diagram that;

Let’s first find length of arc BC.

Expression for length of a circular arc with radius  and angle  rad is;

It is evident from the diagram that;

We can see from the diagram that;

Therefore;

Let us now find BD.

Consider the right-triangle COD in the diagram where  and .

Expression for   trigonometric ratio in right-triangle is;

Therefore;

We have trigonometric ratio;

Now let us find CD.

It is evident from the diagram that;

For OD we again consider the right-triangle COD in the diagram where  and
.

Expression for  trigonometric ratio in right-triangle is;

Therefore;

We have trigonometric ratio;

Therefore, we can write;

Finally, we can find the perimeter of the shaded region;

ii.
 

It is evident from the diagram that;

Let us first find area of sector OCB.

Expression for area of a circular sector with radius  and angle  rad is;

It is evident from the diagram that;

It is evident from the diagram  and ;

Now we find area of triangle COD.

Expression for the area of the triangle is;

It is evident from the diagram that;

From (i) we have found that;

Therefore;

Now we can find area of shaded region;

We are given that  and ;