sides AB and AC and median AD of triangle ABC are respectively proportional to sided PQ and PROBABLY and median PM of another triangle PQR.Show that triangle ABC is similar to triangle PQR?
Answers
Answer:
By Yoshana Chaudhary
Step-by-step explanation:
Given: △ABC & △PQR
AD is the median of △ABC
PM is the median of △PQR
PQ
AB
=
PR
AC
=
PM
AD
→1
To prove: △ABC∼△PQR
Proof:Let us extend AD to point D such that that AD=DE and PM upto point L such that PM=ML
Join B to E, C to E, & Q to L and R to L
(Image 2)
We know that medians is the bisector of opposite side
Hence BD=DC & AD=DE *By construction)
Hence in quadrilateral ABEC, diagonals AE and BC bisect each other at point D
∴ABEC is a parallelogram
∴AC=BE & AB=EC (opposite sides of a parallelogram are equal) →2
Similarly we can prove that
PQLR is a parallelogram.
PR=QL,PQ=LR (opposite sides of a parallelogram are equal) →3
Given that
PQ
AB
=
PR
AC
=
PM
AD
(frim 1)
⇒
PQ
AB
=
QL
BE
=
PM
AD
(from 2 and 3)
⇒
PQ
AB
=
QL
BE
=
2PM
2AD
⇒
PQ
AB
=
QL
BE
=
PL
AE
(As AD=DE,AE=AD+DE=AD+AD=2AD & PM=ML,PL=PM+ML=PM+PM=2PM)
∴△ABE∼△PQL (By SSS similarity criteria)
We know that corresponding angles of similar triangles are equal
∴∠BAE=∠QPL→4
Similarly we can prove that
△AEC∼△PLR
We know that corresponding angles of similar triangles are equal
∠CAE=∠RPL→5
Adding 4 and 5, we get
∠BAE+∠CAE=∠QPL+∠RPL
⇒∠CAB=∠RPQ→6
In △ABC and △PQR
PQ
AB
=
PR
AC
(from 1)
∠CAB=∠RPQ (from 6)
∴△ABC∼△PQR (By SAS similarity criteria)
Hence, proved
Answer:
Given two triangles. ΔABC and ΔPQR in which AB, BC and median AD of ΔABC are proportional to sides PQ, QR and median PM of ΔPQR
AB/PQ = BC/QR = AD/PM
To Prove: ΔABC ~ ΔPQR
Proof: AB/PQ = BC/QR = AD/PM
AB/PQ = BC/QR = AD/PM (D is the mid-point of BC. M is the mid point of QR)
ΔABD ~ ΔPQM [SSS similarity criterion]
Therefore, ∠ABD = ∠PQM [Corresponding angles of two similar triangles are equal]
∠ABC = ∠PQR
In ΔABC and ΔPQR
AB/PQ = BC/QR ———(i)
∠ABC = ∠PQR ——-(ii)
From above equation (i) and (ii), we get
ΔABC ~ ΔPQR [By SAS similarity criterion]
Hence Proved
HOPE IT HELPS U