Q. Given :
Sides AB and BC and median AD of ΔABC are respectively
proportional to sides PQ and QR and median PM of ΔPQR.
Show that : ΔABC ~ ΔPQR.
A
B
C
D
P
Q
R
M
Soln.
...(i)
........ [given]
BC
=
2BD
...(ii)
........ [AD is the median]
QR
=
2QM
...(iii)
........ [PM is the median]
AB
PQ
=
2BD
2QM
=
AD
PM
...(iv)
.... from (i) (ii) & (iii)
∴
median AD
What is a median ??
A line segment drawn from a
vertex of a triangle to the midpoint
of its opposite side.
Sides AB and BC and median AD of ΔABC
are
proportional
median PM
to sides PQ and QR and median PM of ΔPQR
i.e.
=
=
AB
BC
AD
PQ
QR
PM
AB
BC
AD
PQ
QR
PM
Show that : ΔABC ~ ΔPQR
We have one pair of elements
to prove ΔABC ~ ΔPQR
Lets find another pair
AB
PQ
=
BC
QR
=
AD
PM
Do AB, BD and AD
form one triangle ??
ΔABD
Do PQ, QM and PM
form one triangle ??
ΔPQM
EX.6.3 (Q.14)
In ΔABD & ΔPQM,
.... from (iv)
ΔABD
~
ΔPQM
.... [by SSS similarity criterion]
∠B
=
∠Q
.... [corresponding angles of similar
triangles]
...(v)
In ΔABC & ΔPQR
.... from (i)
∠B
=
∠Q
.... from (v)
ΔABC ~ ΔPQR
∴
.... [by SAS similarity criterion]
A
B
C
D
P
Q
R
AB
PQ
=
BD
QM
=
AD
PM
AB
PQ
=
BD
QM
=
AD
PM
S
S
S
M
Triangles are similar
by which criterion ??
SSS criterion
AB
PQ
=
BC
QR
AB
PQ
=
BC
QR
S
A
S
Triangles are similar
by which criterion ??
SAS criterion
EX.6.3 (Q.14)