A triangle $T$ has sides of length $3$, $4$ and $5$.

  (i)     Find the radius of the circumscribed circle of $T$ .

  (ii)     Find the radius of the inscribed circle of $T$ .

A triangle $U$ has sides of length $4$, $5$ and $7$.

  (i)     Show that the orthocentre, H, of $U$ lies outside the triangle.

  (ii)     Show that the foot of the perpendicular from H to the longest side divides it in the ratio $29:20$.

(i)     $T$ is a right angled triangle $\Rightarrow$ the hypotenuse is a diameter     (M1)

circumradius $= 2.5$     A1

(ii)     diagram seen with some sensible unknown(s) given     (A1)

need to solve $3 - r + 4 - r = 5$ (or equivalent)     M1A1

$r = 1$     A1

[6 marks]

(i)     recognition that ${7^2} > {4^2} + {5^2}$     M1  

therefore one of the angles is obtuse     R1

so the orthocentre, H, of $U$ lies outside of the triangle     AG

(ii)     foot of perpendicular from H to longest side is the foot of the perpendicular from A to the longest side     (R1)

EITHER

attempt to solve ${4^2} - {x^2} = {5^2} - {(7 - x)^2}$ or ${4^2} - {(7 - y)^2} = {5^2} - {y^2}$     M1A1

obtain $x = \frac{{20}}{7}$ or $y = \frac{{29}}{7}$     A1

OR

if $\hat B$ is the smallest angle

$\cos \hat B = \frac{{25 + 49 - 16}}{{2 \times 5 \times 7}}$     M1

$= \frac{{58}}{{70}} = \frac{{29}}{{35}}$

$y = 5 \times \frac{{29}}{{35}} = \frac{{29}}{7}$     A1

$x = 7 - \frac{{29}}{7} = \frac{{20}}{7}$     A1

THEN

ratio $29:20$ (accept $20:29$)     AG

Note: Accept the use of Stewart’s theorem.

[6 marks]

A few fully correct answers were seen to this question, but many candidates were unable to make much progress after part a) (i) and a significant minority made no attempt at all. A few fully correct answers were seen to part a) (ii) and part b) (i). In both part a) (ii) and part b) (ii) a majority of candidates were unable to draw a meaningful diagram to enable them to start the question.

A few fully correct answers were seen to this question, but many candidates were unable to make much progress after part a) (i) and a significant minority made no attempt at all. A few fully correct answers were seen to part a) (ii) and part b) (i). In both part a) (ii) and part b) (ii) a majority of candidates were unable to draw a meaningful diagram to enable them to start the question.