# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | May-Jun | (P1-9709/12) | Q#5

Question

In the diagram, triangle ABC is right-angled at C and M is the mid-point of BC. It is given that angle    radians and angle  radians. Denoting the lengths of BM and MC by x,

i.       find AM in terms of x,

ii.       show that

.Solution

i.

We are required to find AM.

Consider right angled triangle AMC.

Pythagorean Theorem

For right angled triangle AMC;

Now we need to find .

Consider right angled triangle ABC.

Expression for  trigonometric ratio in right-triangle is;

For right angled triangle ABC;

Hence;

ii.

It is evident from the diagram that;

Let’s first find .

Consider right angled triangle ABC.

Sum of all the interior angles of triangle equals  radians or .

For right angled triangle ABC;

To find  we next need to find .

Consider right angled triangle AMC.

Expression for  trigonometric ratio in right-triangle is;

For right angled triangle AMC;

From (i) we have ;

Finally, now we can find ;