Answer
451.8k+ views
Hint: Let’s make use of the property of congruence of triangles & analyse the sides of triangles by medians to approach the solution.
Complete step-by-step answer:
Given that $\vartriangle ABC \sim \vartriangle PQR$
So, from the property of congruence triangles, we can write
$\dfrac{{AB}}{{PQ}} = \dfrac{{BC}}{{QR}} = \dfrac{{AC}}{{PR}}$ (Corresponding sides of congruent triangles)
Also, $\angle A = \angle P,\angle B = \angle Q,\angle C = \angle R$ (Corresponding angles of congruent triangles)
Given that AD and PM are medians of the triangle ABC and PQR respectively
If a median is drawn from a vertex to a side, then it divides the side equally
So, from this we can write BD=CD$ \Rightarrow BC = 2BD$
QM=MR$ \Rightarrow $QR=2QM
Hence, from this we can write
$\dfrac{{AB}}{{PQ}} = \dfrac{{2BD}}{{2QM}} = \dfrac{{AC}}{{PR}}$
$ \Rightarrow \dfrac{{AB}}{{PQ}} = \dfrac{{BD}}{{QM}} = \dfrac{{AC}}{{PR}}$
So, now we can write in $\vartriangle ABD$ and $\vartriangle PQM$
$\dfrac{{AB}}{{PQ}} = \dfrac{{BD}}{{QM}}$
and $\angle B = \angle Q$
And so , from SAS similarity, we can write
$\vartriangle ABD \sim \vartriangle PQM$
$\therefore \dfrac{{AB}}{{PQ}} = \dfrac{{AD}}{{PM}}$ (Congruent sides of congruent triangles)
So, this is what we had to prove, hence the result is proved.
Note: When solving these types of problems first prove the congruence of the triangles and from this find out the ratio of the corresponding sides in accordance to the RHS which has to be proved.
Complete step-by-step answer:
![seo images](https://www.vedantu.com/question-sets/ed986269-c51a-4fdf-91db-610b29117bbc1463940698951809872.png)
Given that $\vartriangle ABC \sim \vartriangle PQR$
So, from the property of congruence triangles, we can write
$\dfrac{{AB}}{{PQ}} = \dfrac{{BC}}{{QR}} = \dfrac{{AC}}{{PR}}$ (Corresponding sides of congruent triangles)
Also, $\angle A = \angle P,\angle B = \angle Q,\angle C = \angle R$ (Corresponding angles of congruent triangles)
Given that AD and PM are medians of the triangle ABC and PQR respectively
If a median is drawn from a vertex to a side, then it divides the side equally
So, from this we can write BD=CD$ \Rightarrow BC = 2BD$
QM=MR$ \Rightarrow $QR=2QM
Hence, from this we can write
$\dfrac{{AB}}{{PQ}} = \dfrac{{2BD}}{{2QM}} = \dfrac{{AC}}{{PR}}$
$ \Rightarrow \dfrac{{AB}}{{PQ}} = \dfrac{{BD}}{{QM}} = \dfrac{{AC}}{{PR}}$
So, now we can write in $\vartriangle ABD$ and $\vartriangle PQM$
$\dfrac{{AB}}{{PQ}} = \dfrac{{BD}}{{QM}}$
and $\angle B = \angle Q$
And so , from SAS similarity, we can write
$\vartriangle ABD \sim \vartriangle PQM$
$\therefore \dfrac{{AB}}{{PQ}} = \dfrac{{AD}}{{PM}}$ (Congruent sides of congruent triangles)
So, this is what we had to prove, hence the result is proved.
Note: When solving these types of problems first prove the congruence of the triangles and from this find out the ratio of the corresponding sides in accordance to the RHS which has to be proved.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)