Answer
Verified
429.3k+ views
Hint: In order to solve this question, we have to apply similarity rules of triangles and in which side and angles helps us to show the similarities of these triangles.
Complete step-by-step answer:
According to given question,
$\dfrac{{AB}}{{PQ}} = \dfrac{{BC}}{{QR}} = \dfrac{{AD}}{{PM}} - - - - - \left( 1 \right)$
In $\Delta ABC$, since $AD$ is the median,
$BD = CD = \dfrac{1}{2}BC$
Or $BC = 2BD - - - - - \left( 2 \right)$
Similarly, $PM$ is the median,
$QM = RM = \dfrac{1}{2}QR$
Or $QR = 2QM - - - - - \left( 3 \right)$
Substituting the value of $BC,QR$ in equation (1), we get
$\dfrac{{AB}}{{PQ}} = \dfrac{{2BD}}{{2QM}} = \dfrac{{AD}}{{PM}}$
$\dfrac{{AB}}{{PQ}} = \dfrac{{BD}}{{QM}} = \dfrac{{AD}}{{PM}} - - - - - \left( 4 \right)$
Since all three sides are proportional.
Therefore, by SSS similarity rule,
$\Delta ABD \sim \Delta PQM$
Hence,$\angle B = \angle Q - - - - - - \left( 5 \right)$,
corresponding angles of similar triangles are equal.
In $\Delta ABC\& \Delta PQR$
Using (5), we get
$\angle B = \angle Q$
Given, $\dfrac{{AB}}{{PQ}} = \dfrac{{BC}}{{QR}}$
Hence by SAS Similarity of triangle.
$\Delta ABC \sim \Delta PQR$
Note: Whenever we face these types of questions the key concept is that we have to take small triangles and by similarity rules show them similar by which we get two sides or one sides and one angle equality and we will easily get our desired answer.
Complete step-by-step answer:
According to given question,
$\dfrac{{AB}}{{PQ}} = \dfrac{{BC}}{{QR}} = \dfrac{{AD}}{{PM}} - - - - - \left( 1 \right)$
In $\Delta ABC$, since $AD$ is the median,
$BD = CD = \dfrac{1}{2}BC$
Or $BC = 2BD - - - - - \left( 2 \right)$
Similarly, $PM$ is the median,
$QM = RM = \dfrac{1}{2}QR$
Or $QR = 2QM - - - - - \left( 3 \right)$
Substituting the value of $BC,QR$ in equation (1), we get
$\dfrac{{AB}}{{PQ}} = \dfrac{{2BD}}{{2QM}} = \dfrac{{AD}}{{PM}}$
$\dfrac{{AB}}{{PQ}} = \dfrac{{BD}}{{QM}} = \dfrac{{AD}}{{PM}} - - - - - \left( 4 \right)$
Since all three sides are proportional.
Therefore, by SSS similarity rule,
$\Delta ABD \sim \Delta PQM$
Hence,$\angle B = \angle Q - - - - - - \left( 5 \right)$,
corresponding angles of similar triangles are equal.
In $\Delta ABC\& \Delta PQR$
Using (5), we get
$\angle B = \angle Q$
Given, $\dfrac{{AB}}{{PQ}} = \dfrac{{BC}}{{QR}}$
Hence by SAS Similarity of triangle.
$\Delta ABC \sim \Delta PQR$
Note: Whenever we face these types of questions the key concept is that we have to take small triangles and by similarity rules show them similar by which we get two sides or one sides and one angle equality and we will easily get our desired answer.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Fill the blanks with proper collective nouns 1 A of class 10 english CBSE
What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.
One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Net gain of ATP in glycolysis a 6 b 2 c 4 d 8 class 11 biology CBSE
What organs are located on the left side of your body class 11 biology CBSE