untitled

QonABand AC respectively so that AP = DE

andAQ = DF. Join PandQ.

Proof:

Steps Justificaltin

(1) In 'APQ and 'DEF ,AP DE, AQ DF

and included ‘A^ included‘D
?'ABC#'DEF
?‘A ‘D,‘APQ ‘E,‘AQP ‘F.
(2) Again,

since DF

AC

DE

AB

, so AQ

AC

AP

AB

?PQllBC

Therefore,‘ABC^ ‘APQ ‘ACB^ ‘AQP

? ‘A ‘D,‘B ‘E ‘C ‘F

i.e., triangles ABC and

and

and

DEF are equiangular.

Therefore 'ABC and 'DEF are similar.

[SAS Theorem]

[Theorem 2]

Theorem 8

The ratio of the areas of two similar triangles is equal to the ratio of squares on

any two matching sides.

Proposition : Let the triangles ABC and DEF

be similar and BC and EF be their matching

sides respectively. It is required to prove that

'ABCt'DEF BC^2 tEF^2

Construction:Draw perpendiculars AG and

DH on BC and EF respectively. Let AG h,

DH p.

Proof:

Steps Justificaltin

(1) ABC BCh DEF EF.p

2

1

.

2

1

' '

pEF

hBC

EFp

BCh

DEF

ABC

.

.

.

.

2

1

2

1

'

'

? =

EF

BC

p

h

u

(1) But in the triangles ABG and DEG,

E,

B

‘

‘

‘AGB ‘DHE (= 1 right angle)

and