ABCDEFGH IS AN OCTAGON IN WHICH ALL EIGHT ANGLES ARE EQUAL. IF...

2. ABCDEFGH is an octagon in which all eight angles are equal.

If AB = 7, BC = 4, CD = 2, DE = 5, EF = 6 and FG = 2, determine the sum of

the lengths of GH and HA .

F E

G

D

C

H

【Solution】

A B

Extend AB , CD , EF and GH to form a quadrilateral

R Q

PQRS . Each angle of ABCDEFGH is 135 ◦ _{. Hence}

2 6

5

each of PCB , QED , RGF and SAH are 45 ◦ ₋₄₅ ◦ ₋₉₀ ◦

triangles. It follows that PQRS is a rectangle. We have

2

= 2

2 =2 2 , QE = QD = 5 5 2

PB = PC = 4

4

2 and

7

S

A B P

RG = RF = 2

2 = 2 .

Hence

5 2 3 2

SH = SA = SP − AB − BP = RQ − AB − BP = + + − − = − so

( 2 6 7 2 2) 1

2 2

that HA=3 − 2 . Also,

(2 2 2 ) 2 ( 1) 3 2 2

GH = RS − RG − SH = QP − RG − SH = + + − − − = +

It follows that GH + HA = + 6 2

Ans: 6 + 2