Question

Seg PA,Seg QB and RC are perpendicular Seg AC
PA=x,QB=y,RC=z
Prove 1÷x+1÷z=1÷you

Answer 1

Step-by-step explanation:

Consider △CAP and △CBQ

∠CAP=∠CBQ=90

∘

∠PCA=∠QCB (common angle)

So by AA, △CAP∼△CBQ

Hence

AP

BQ

=

AC

BC

x

y

=

AC

BC

(1)

Now consider △ACR and △ABQ

∠ACR=∠ABQ=90

∘

∠QAB=∠RAC (common angle)

So by AA, △ACR∼△ABQ

Hence

CR

BQ

=

AC

AB

z

y

=

AC

AB

(2)

Adding (1) and (2)

x

y

+

z

y

=

AC

BC

+

AC

AB

y(

x

1

+

z

1

)=

AC

BC+AB

y(

x

1

+

z

1

)=

AC

AC

y(

x

1

+

z

1

)=1

x

1

+

z

1

=

y

1

Hence proved.