# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2005 | Oct-Nov | (P1-9709/01) | Q#3

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Question

In the diagram, ABED is a trapezium with right angles at E and D, and CED is a straight line.
The lengths of AB and BC are
and  respectively, and angles BAD and CBE are and  respectively.

i.
Find the length of CD in terms of .

ii.       Show that angle CAD =

Solution

From the given information we can compile following data;

i.

It is evident from the diagram that;

We can find  from ;

Expression for   trigonometric ratio in right-triangle is;

From the given diagram for ;

We can find  from ;

Expression for   trigonometric ratio in right-triangle is;

From the given diagram for ;

It is evident from the diagram that;

Hence;

Now we can substitute the values in equation;

ii.

Expression for  trigonometric ratio in right-triangle is;

It is evident from the diagram that for ;

From (i) we know that;

To find , it is evident from the diagram that;

It is also evident from the diagram that;

Therefore;

Now we find out  .

Expression for  trigonometric ratio in right-triangle is;

 For For From From

Now we can substitute the values in the equation;

Finally we have the values to substitute in equation;