Everything Maths Grade 11

CHAPTER 16. GEOMETRY 16.4

Given:�ABC withAˆ = 90◦

1

2

A

B C

D

2

1

R.T.P.: BC^2 = AB^2 + AC^2

Proof:

LetCˆ = x

∴ DACˆ = 90◦− x (∠’s of a� )

∴ DABˆ = x

ABDˆ = 90◦− x (∠’s of a� )

BDAˆ = CDAˆ =Aˆ = 90◦

∴�ABD|||�CBA and�CAD|||�CBA (AAA)

∴

AB

CB

=

BD

BA

=

�

AD

CA

�

and

CA

CB

=

CD

CA

=

�

AD

BA

�

∴ AB^2 = CB× BD and AC^2 = CB× CD

∴ AB^2 + AC^2 = CB(BD + CD)

= CB(CB)

= CB^2

i.e. BC^2 = AB^2 + AC^2