1.15: Right triangle with surds

0606/13/M/J/19 — Question 7 · 8 marks

Do not use a calculator in this question.

1.15 diagram

In this question, all lengths are in centimetres.

The diagram shows the triangle

ABC

such that

AB = 2\sqrt{5} - 1

,

BC = 2 + \sqrt{5}

and angle

ABC = 90\degree

.

(i)

Find the exact length of

AC

.

[3]

(ii)

Find

\tan ACB

, giving your answer in the form

p + q\sqrt{r}

, where

p

,

q

and

r

are integers.

[3]

(iii)

Hence find

\sec^{2} ACB

, giving your answer in the form

s + t\sqrt{u}

where

s

,

t

and

u

are integers.

[2]