Everything Maths Grade 11

16.4 CHAPTER 16. GEOMETRY

Proof: In�’s AGH and DEF

AG = DE; AH = DF (constant)

Aˆ =Dˆ (given)

∴�AGH≡�DEF (SAS)

∴ AGHˆ =Eˆ =Bˆ

∴ GH� BC (corresponding∠’s equal)

∴

AG

AB

=

AH

AC

(proportion theorem)

∴

DE

AB

=

DF

AC

(AG = DE; AH = DF)

Tip ∴�ABC|||�DEF

||| means “is similar to”

Theorem 4. Similarity Theorem 2:Triangles with sides in proportion are equiangular and therefore

similar.

A

B C

D E

h 1

h 2

Given:�ABC with line DE such that

AD

DB

=

AE

EC

R.T.P.: DE� BC;�ADE|||�ABC

Proof:

Draw h 1 from E perpendicular to AD, and h 2 from D perpendicular to AE.

Draw BE and CD.

area�ADE

area�BDE

=

1

2 AD. h^1

1

2 DB. h^1

=

AD

DB

area�ADE

area�CED

=

1

2 AE. h^2

1

2 EC. h^2

=

AE

EC

but

AD

DB

=

AE

EC

(given)

∴

area�ADE

area�BDE

=

area�ADE

area�CED

∴ area�BDE = area�CED

∴ DE� BC (same side of equal base DE, same area)

∴ ADEˆ = ABCˆ (corresponding∠’s)

and AEDˆ = ACBˆ

∴�ADE and�ABC are equiangular

∴�ADE|||�ABC (AAA)

Theorem 5. Pythagoras’ Theorem:The square on the hypotenuse of a right angledtriangle is equal to

the sum of the squares on the other two sides.