# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Oct-Nov | (P1-9709/12) | Q#8

Question The diagram shows an isosceles triangle ACB in which AB = BC = 8 cm and AC = 12 cm. The arc  XC is part of a circle with centre A and radius 12 cm, and the arc YC is part of a circle with centre B  and radius 8 cm. The points A, B, X and Y lie on a straight line.

i.       Show that angle CBY = 1.445 radians, correct to 4 significant figures.

ii.       Find the perimeter of the shaded region.

Solution

i.

We are required to find the angle .

We are given that points A, B, X and Y lie on a straight line, therefore, .

It is evident from the diagram that;  As we have found above that ; Hence, if we find .
We can find .

To find let’s consider . We are given that it is an isosceles triangle with;  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;   We have lengths of all three sides of the triangle; therefore, we can find the angle between the two sides and .          Hence;   ii.

It is evident from the diagram that; First, lets find length of arc CY.

Expression for length of a circular arc with radius and angle rad is; For the arc CY;   Secondly, we find length of arc CX.

Expression for length of a circular arc with radius and angle rad is; For the arc CX;  We need value of . It is evident from the diagram that .

To find let’s consider . We are given that it is an isosceles triangle with;  Sum of all the three interior angles of a triangle is or rad.

Therefore; Angles opposite to the equal sides of an isosceles triangle are also equal.

Since AB=BC; Hence;  We have found above that; Hence;    Hence; Now we can find;  Lastly, we need length XY.

It is evident from the diagram that;  We are given that the arc XC is part of a circle with centre A and radius 12 cm. It is evident from the diagram that;  It is also given that ; therefore,  We are given that the arc YC is part of a circle with centre B and radius 8 cm.. Hence;   Now we can find;      