side ab and AC and median ad of a triangle ABC are respectively proportional to sides PQ and PR and medium PM of another triangle pqr so that triangle ABC similar Triangle pqr
Answers
Step-by-step explanation:
Given that triangle ABC and ΔPQR in which AD and PM are medians drawn on sides BC and QR respectively.
It is given that AB/PQ = AC/PR = AD/PM
We have to prove that ΔABC ~ ΔPQR
Construction: Produce AD to E such that AD = DE and PM to F such that PM = MF
From the figure,
In ΔABD and ΔCDE,
AD = DE [by construction]
∠ADB = ∠CDE [vertically opposite angles]
BD = DC [Since AD is a median]
So, by SAS congruent condition
ΔABC ≅ ΔPQR
AB = CE [by CPCT]
Similarly, we can prove
ΔPQM ≅ ΔRMF
PQ = RF [by CPCT]
Now, given that
AB/PQ = AC/PR = AD/PM
CE/RF = AC/PR = 2AD/2PM
CE/RF = AC/PR = AE/PF [Since AE = AD + DE and AD = DE, Same for PF]
By using SSS Congruent condition.
ΔACE ≅ ΔPRF
=> ∠1 = ∠2 ......1
Similarly, ∠3 = ∠4 ......2
Adding equations 1 and 2, we get
∠1 + ∠3 = ∠2 + ∠4
=> ∠A = ∠P
Now, in ΔABC and ΔPQR
AB/PQ = AC/PR
and ∠A = ∠P
By SAS similar condtion,
ΔABC ~ ΔPQR
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