the ratio of the areas of two similiar triangles is equal to square of ratio of their corresponding sides. Prove it.
Answers
Answer:
If two triangles are similar, then the ratio of the area of both triangles is proportional to the square of the ratio of their corresponding sides. This proves that the ratio of the area of two similar triangles is proportional to the squares of the corresponding sides of both the triangles.
Step-by-step explanation:
Let the two triangles be:
ΔABC and ΔPQR
Area of ΔABC=
2
1
×BC×AM……………..(1)
Area of ΔPQR=
2
1
×QR×PN……………………..(2)
Dividing (1) by (2)
ar(PQR)
ar(ABC)
=
2
1
×QR×PN
2
1
×BC×AM
ar(PQR)
ar(ABC)
=
QR×PN
BC×AM
…………………..(1)
In ΔABM and ΔPQN
∠B=∠Q (Angles of similar triangles)
∠M=∠N (Both 90
∘
)
Therefore, ΔABM∼ΔPQN
So,
AM
AB
=
PN
PQ
…………………….(2)
From 1 and 2
ar(PQR)
ar(ABC)
=
QR
BC
×
PN
AM
⇒
ar(PQR)
ar(ABC)
=
QR
BC
×
PQ
AB
…………………..(3)
PQ
AB
=
QR
BC
=
PR
AC
………….(ΔABC∼ΔPQR)
Putting in ( 3 )
ar(PQR)
ar(ABC)
=
PQ
AB
×
PQ
AB
=(
PQ
AB
)
2
⇒
ar(PQR)
ar(ABC)
=(
PQ
AB
)
2
=(
QR
BC
)
2
=(
PR
AC
)
2
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