In the given figure PA, QB and RC are each perpendiculars to AC. Prove that
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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.
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