If ∆PQR ~ ∆ABC such that PQ : AB = 3 : 4 then find the ratio of their medians.
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Step-by-step explanation:
ANSWER
Consider the triangles △ABC and △PQR
AD and PM being the mediums from vertex A and P respectively.
Given : △ABC∼△PQR
To prove :
PQ
AB
=
PM
AD
It is given that △ABC∼△PQR
⇒
PQ
AB
=
QR
BC
=
PR
AC
[ from the side-ratio property of similar △ s]
⇒∠A=∠P,∠B=∠Q,∠C=∠R.......(A)
BC=2BD;QR=2 QM [P,M being the mid points of BC q QR respectively]
⇒
PQ
AB
=
2QM
2BD
=
PR
AC
⇒
PQ
AB
=
QM
BD
=
PR
AC
........(1)
Now in △ABDq△PQM
PQ
AB
=
QM
BP
........[ from (1)]
∠B=∠Q........[ from (A)]
⇒△ABD∼△PQM [ By SAS property of similar △ s] from the side property of similar △ s Hence proved
PQ
AB
=
PM
AD
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