If the areas of two similar triangles are equal then prove
that they are congruent.
Q.
Given :
ΔABC ~ ΔPQR
ar (ΔABC) = ar (ΔPQR)
To Prove :
ΔABC
≅
ΔPQR
Proof.
... [given]
A
B
C
R
Q
P
ar
(ΔABC)
ar
(ΔPQR)
∴
AB2
PQ2
=
BC2
QR2
=
AC2
PR2
=
[i]
The ratio of the areas of two
similar triangles is equal to the
ratio of the squares of the
corresponding sides
...
But
[ii]
... [given]
∴
=
1
1
... from [i] & [ii]
If the areas of two similar triangles are equal
then prove
that they are congruent.
ΔABC ~ ΔPQR
ar (ΔABC) = ar (ΔPQR)
ar
(ΔABC)
ar
(ΔPQR)
=
1
∴
AB2
PQ2
BC2
QR2
=
AC2
PR2
=
EX.6.4 (Q.4)
∴
AB²
=
PQ²
,
BC²
=
QR²
,
AC²
=
PR²
∴
AB
=
PQ
,
BC
=
QR
,
AC
=
PR
... [by SSS congruency]
∴
ΔABC
≅
ΔPQR
∴
=
,
=
,
=
AB2
PQ2
BC2
QR2
=
AC2
PR2
=
=
1
AB2
PQ2
1
BC2
QR2
1
AC2
PR2
1
A
B
C
R
Q
P
Taking square roots
Triangles are congruent by which test ??
S
S
S
S
S
S
SSS
EX.6.4 (Q.4)