# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Oct-Nov | (P1-9709/12) | Q#6

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 ;          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;    