# 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

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**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 ;

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