WAEC 2009 · Paper 2 · Q5

(a)
In the diagram, the two circles intersect at $X$ and $Y$ . The centre $O$ of the smaller circle is on the circumference of the bigger circle. $A$ and $B$ are any two points on the major arcs, one on each circle. Find an equation connecting $a$ and $b$ .
Model answer
Join $OX$ and $OY$ . In the small circle, $\angle XOY = 2b^\circ$ (angle at the centre). $AXOY$ is a cyclic quadrilateral of the big circle, so $a + 2b = 180$ .
(b)
In the diagram, $\angle QPR = \angle PTR = 90^\circ$ , $|PR| = 8$ cm and $|QP| = 6$ cm. Find $|TR|$ .

Worked solution (try it first)

Join $OX$ and $OY$ .
In the smaller circle (centre $O$ ), the angle at the centre is twice the angle at the circumference: $\angle XOY = 2b^\circ$ .
$A$ , $X$ , $O$ and $Y$ all lie on the bigger circle, so $AXOY$ is a cyclic quadrilateral and $\angle XAY + \angle XOY = 180^\circ$ .
So $a + 2b = 180$ .