 Hello friends, welcome to the session. I am Alka. Let's discuss triangles. Our given question is side AB and AC and median AD of a triangle ABC are respectively proportional to size PQ and PR and median PM of another triangle PQR Show that triangle ABC is similar to triangle PQR Here is the figure according to the question that is triangle ABC and triangle PQR Where AD is the median of triangle ABC and PM is the median of triangle PQR And we have done the construction. We have drawn the line DE parallel to AB and align MN parallel to PQ. Now let's start with the solution Then let's start with the solution We are given to Triangles triangle C and triangle PQR We are also given that AD is the Median angle AB CM is the median of Triangle PQ now We are also given that AB upon PQ equal to AC upon PR equal to AD upon PM. Since we are given that the sides AB and AC and median AD of triangle ABC Are proportional to the sides PQ and PR and median PM of another triangle PQR Now we have to prove that Triangle AB C is similar to triangle PQ We have done the construction In the triangles that is we have drawn the E parallel to AB and MN parallel to PQ draw DE parallel to AB and parallel to PQ Let's see the proof Now since we all know that by mid value midpoint theorem That the line drawn through the midpoint of one side of a triangle parallel to the other side Biceps the third side that is we have drawn line DE From the midpoint of the VC and this line DE is parallel to AC So this line DE bisects the third side which is AC that is midpoint theorem So by midpoint theorem we have AC equal to twice of AE and equal to twice of PM Let this PR first equation and We can also say that DE equal to half of AB DE is half of AB and PQ by Mid so let this be our second equation now We are given that AB PQ equal to AC We are given now AB can be written as 2 DE upon written as 2 MN By using our second equation. We see that AB equal to 2 DE and PQ equal to 2 MN equal to AC is 2 Ae that is equal to 2 PM using our equation first, which is equal to AD upon PM We see that 2 2 cancel out we get DE upon MN equal to AE upon PN equal to AD upon PM Therefore, we can say that Triangle AD E is similar to triangle PM N by SSS criteria of similarity Now since the triangle AD is similar to triangle PM N Therefore the corresponding angles are equal. We can say that angle D A E is equal to angle M P N VAE VAE is equal to angle M P N Since we know the triangle AD E is similar to triangle PM N Now this can also be written as VAC Equal to angle M P R You can see from the figure that angle D AC Equal to angle M P Let this be our third equation Now similarly we can say that D A V equal to angle M P Q We can see from the figure that angle D A V Equal to angle M P Q Let this be our fourth equation now on adding Equation third and fourth We get angle D AC plus angle D AB D AC plus D AB equal to angle VAC So we can write angle V AC Equal to now we ran M P R plus M P Q M P R plus M P Q equal to Q P R So this is angle Q Q P this we are fifth equation Now We can see in triangle A B C triangle P Q R Let angle B AC Equal to angle Q P R from our equation fifth AB upon P Q Equal to AC upon Given to us Now we can say that Therefore criteria we have Triangle A B C is similar to triangle P Q If you understood the solution, goodbye and take care