Math, asked by rahi01435, 7 months ago

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

Answered by yoshanachaudhary

Answer:

By Yoshana Chaudhary

Step-by-step explanation:

Given: △ABC & △PQR

AD is the median of △ABC

PM is the median of △PQR

→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

(frim 1)

⇒

(from 2 and 3)

⇒

2PM

2AD

⇒

(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

(from 1)

∠CAB=∠RPQ (from 6)

∴△ABC∼△PQR (By SAS similarity criteria)

Hence, proved

Answered by psupriya789

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

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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?​

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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?