3.1: Trapezium with surds: perimeter, area, tan and sec

0606/12/F/M/21 — Question 7 · 9 marks

Do not use a calculator in this question.

In this question all lengths are in centimetres.

The diagram shows a trapezium

ABCDE

such that

AB

is parallel to

EC

and

ABCD

is a rectangle. It is given that

BC = \sqrt{17} + 1

ED = \sqrt{17} - 1

and

DC = \sqrt{17} + 4

(a)

Find the perimeter of the trapezium, giving your answer in the form

a + b\sqrt{17}

, where

a

and

b

are integers.

[3]

(b)

Find the area of the trapezium, giving your answer in the form

c + d\sqrt{17}

, where

c

and

d

are integers.

[2]

(c)

Find

\mathrm{tan}\,AED

, giving your answer in the form

\dfrac{e + f\sqrt{17}}{8}

, where

e

and

f

are integers.

[2]

(d)

Hence show that

\mathrm{sec}^{2} AED = \dfrac{81 + 9\sqrt{17}}{32}

[2]