Question

The diagram shows a metal plate ABCD made from two parts. The part BCD is a semicircle. The part DAB is a segment of a circle with centre O and radius 10 cm. Angle BOD is 1.2 radians.

    i.       Show that the radius of the semicircle is 5.646 cm, correct to 3 decimal places.

   ii.       Find the perimeter of the metal plate.

  iii.       Find the area of the metal plate.

Solution

i.
 

We are given that BCD is a semicircle. This leads to the fact that BD is diameter of the semicircle BCD.

It is evident from the diagram that;

Now we need to find BD.

Let’s consider . It is evident from the diagram that  is an isosceles triangle with;

We also have the angle included by two equal sides;

We need to find third side of .

If we have lengths of two sides and the included angle, we can use law of cosines to find the 3rd  length of the triangle. Law of cosines is; 

Therefore, for ;

Hence, radius of semicircle BCD;

ii.
 

It is evident from the diagram that;

Let’s first find length of arc DAB.

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

We are given that;

It is evident from the diagram that;

Therefore;

Now let’s find circumference of semicircle BCD.

Therefore;

We have found in (i) that;

Hence;

Finally;

iii.
 

It is evident from the diagram that;

Let’s first find area of sector DAB.

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

We are given that;

It is evident from the diagram that;

Therefore;

Now let’s find area of semicircle BCD.

We have found in (i) that;

Hence;

Now we proceed to find area of  .

Expression for the area of a triangle for which two sides (a and b) and the included angle (C ) is given;

Let’s consider . It is evident from the diagram that  is an isosceles triangle with 

We also have the angle included by two equal sides;

Therefore;

Finally;