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

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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;

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