# If AD and PM are medians of triangles ABC and PQR , respectively where $\vartriangle ABC \sim \vartriangle PQR$ , prove that $\dfrac{{AB}}{{PQ}} = \dfrac{{AD}}{{PM}}$

Answer

Verified

361.8k+ 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:

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.

Last updated date: 24th Sep 2023

•

Total views: 361.8k

•

Views today: 9.61k

Recently Updated Pages

What do you mean by public facilities

Please Write an Essay on Disaster Management

Paragraph on Friendship

Slogan on Noise Pollution

Disadvantages of Advertising

Prepare a Pocket Guide on First Aid for your School

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

The poet says Beauty is heard in Can you hear beauty class 6 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is the past tense of read class 10 english CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE