Q1.  [Maximum mark: 24]

(a) (i) Use trigonometric ratios to prove that in a triangle ABC

                                      $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ .

(ii)   Prove that the area of the triangle is  $\frac{1}{2}ab\sin C$ .

(iii)  Given that R denotes the radius of the circumscribed circle prove that       $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$ .

(iv)   Hence show that the area of the triangle ABC is $\frac{abc}{4R}$ .

(b)   A new triangle DEF is positioned within a circle radius R such that DF is a diameter as shown in the following diagram.

(i)  Find in terms of R, the two values of (DE)² such that the area of the shaded region is twice the area of the triangle DEF.

(ii)   Using two diagrams, explain why there are two values of (DE)² .

(c)   A parallelogram is positioned inside a circle such that all four vertices lie on the circle. Prove that it is a rectangle.

Q2.   [Maximum mark: 14]

(a)   Figure 1 shows a tangent [PQ] at the point Q of a circle and a line [PS] meeting the circle at the points R , S and passing through the centre O of the circle. Show that PQ² = PR × PS .

(b)   Figure 2 shows a triangle ABC inscribed in a circle. The tangents at the points A , B , C meet the opposite sides of the triangle externally at the points D , E , F respectively.

(i)    Show that $\frac{A{D}^2}{B{D}^2}=\frac{CD}{BD}$ .
(ii)   By considering a pair of similar triangles, show that $\frac{AD}{BD}=\frac{AC}{AB}$ and hence that $\frac{CD}{BD}=\frac{A{C}^2}{A{B}^2}$ .
(iii)   By writing down and using two further similar expressions, show that the points D , E , F are collinear.

Q3.  [Maximum mark: 15]

Q4.    [Maximum mark: 15]

Q5.    [Maximum mark: 15]

Q6.    [Maximum mark: 15]