 hello and welcome to another problem solving session on congruent triangles now in this question it's given that d is the midpoint of hypotenuse ac of right angle abc okay so this is right angle triangle abc and d is the midpoint that means clearly ad is equal to dc okay it's given you have to prove that bd this bd is half of ac now on the first appearance it's it's kind of difficult to understand how bd is related to ac but if you have studied circles and you know the properties of circle that in a semi circle the angle subtended is always 90 degree then from that perspective it is just a one-line proof because ba and dc happens to be equal so d can be assumed to be the center of that given circle and b is also b will lie on that semi circle is it so if i draw semi circle like that like that then b is lying on that semi circle and d will be the center so obviously da will be equal to db is equal to dc as the radii but we are not going to take up that truth we are going to take up root of congruence okay so how to go about it how do we prove it using congruence now now one thing is very clear that bd is half ac that means it has to be we have to get two triangles where bd and ac are involved so clearly one triangle or the set of two triangles which is here is this adb and dbc but apart from ad being equal to bc we don't see any relation between bd and the hypotenuse ac okay the only other thing which we know about right angle triangle is ac square is equal to ab square plus bc square which is not going to help much here also it's not known whether bd is perpendicular to ac so hence it is not going to help much here so that means in such circumstances we need to see whether we can do or we can solve this proof or we can do this proof by some construction that's an important aspect of proving right now as i told you if we are going to prove that bd is half ac so they need to be part of two triangles where i find some relation between now obviously the two triangles of which bd is a part clearly is not going to help much here so hence we need to look out of the triangle isn't it so if suppose bd happens to be equal to half of some other side which is which is equal to ac then our purpose is solved so what i'm going to do is i'm going to construct something so what i'm going to do is i'm going to extend bd okay i'm going to extend bd such that bd is equal to let's say de okay so now bd automatically becomes half of bd the moment i prove that be is equal to ac then our job is done that's what we are going to do so and also we are going to join these two points okay so let's now do the proof formally so let's first write what is given so it's given that angle abc is equal to 90 degrees first of all right and and and bd sorry ad is equal to ad is equal to dc this is given right we have to prove that bd is equal to half ac okay now so what is the construction we have done construction we have done this construction what is this bd extended extended to e right why are we doing this it will become clear in just some moments bd extended to be such that bd is equal to de okay and ec is joined so let me write it a little bit more clearly ec is joined very good right so this is clear now what do we achieve from this the moment these two are equal then can you see these two triangles are congruent right the moment that will be congruent what do we achieve we would achieve that these two sides ab and ec will become equal the moment ab and ec are equal so if you see bc is there so in triangle abc and e bc these two triangles ab is equal to ec by proving angle b will be equal to angle c both will be 90 degree will prove that as well and bc is a common side then automatically ac will become ve because these two triangles would be congruent in that case so that's what the line of approach is okay so let's first try and prove right with that prove that adb is congruent to cdb okay so i'm writing in triangles which triangle adb first one and triangle cde these are two triangles okay what do we get here we get ac a sorry ad is equal to dc this is given it's already given right and vd is equal to de by construction so i'm writing by construction i will write in shorthand by construction and angle adb angle adb is equal to angle cde cde and why is this this is because of vertically opposite angles correct therefore what do we conclude therefore therefore triangle adb is congruent to which triangle cde cdb so correspondence of vertices are important so please check the order of the point should be a should be equal to cd should be equal to d and e d should be equal to e that is there okay so once that is established we can say two things one ab is equal to cd for cd cd and this is because of corresponding parts of congruent triangles point number one and and angle b ad let's say this is x and this is also x right cpcd again so b ad is equal to angle b ad is equal to angle ecd ecd this is cpcd again but this also happens to be alternate interior angle check this is ab and ec are two lines whose alternate interior angles are same therefore we can say and i'm writing here let's write let's use the space so we can write ab is parallel to ec right why because angle b ad since angle b ad is equal to angle ecd is equal to x both are same right the moment these are parallel lines what can we say so if ab is parallel to ec then this angle is also going to be equal to this angle yes or no 90 degrees because both of them are parallel lines so co-interior angle will be adding up to 180 degrees so hence abc plus ecb will be 180 degrees why is this co-interior angles okay co-interior angles therefore abc anyways is 90 degree right this is 90 degree so hence angle ecb will be how much 180 degrees minus angle abc minus angle abc is minus 90 degrees which is 90 degrees right so ecb also is 90 degrees so i can say this angle is 90 is what i'm showing it in little enlarged view or let me just do one thing i'm going to reduce this thing and i'm saying so let us remove this and i'm saying this angle is 90 degrees okay now once that is established now let's compare these two triangles which two abc look carefully abc this one and e b c okay so i'm saying in triangle abc abc and triangle ecb look at the order in which i have written the vertices so abc and ecb what is that you can say ab is equal to ce how do i know from here given right so ab is equal to ce then angle abc is equal to angle c sorry angle ecb so i have to write angle ecb correct angle abc is equal to angle ecb both are 90 degrees and what else so let me write here now what else what else and third is bc is equal to cb common side common side therefore conclusion what can we can do we can conclude that triangle abc is congruent to triangle which one ecb ecb check the correspondence a is equal to e b is equal to c 90 degrees and angle c is equal to angle b right this is uh correct order of vertices therefore hence we conclude hence we can conclude what ac ac is equal to be right and why is this this is cpct corresponding parts of corresponding triangle so ac is equal to be that means bd is nothing but ac is equal to 2 bd why because if you see bd was equal to be from here check right so hence ac is equal to 2 bd so hence bd will be equal to half ac right that's what we wanted to establish and we have successfully established it okay so we understood the proof appeared to be a little longer there are two times we are proving that two different set of triangles are congruent and using that property we have achieved the result so the only thing you have to keep in mind is when it is nothing you know nothing is appearing to be you know giving you the desired result so you would like to try construction and construction in such a way that the desired result is part of two triangles right so here in this case bd and ac we constructed two triangles where bd was part of one triangle ac was part of another triangle and we established some congruence between the two those two triangles right and hence we were able to solve this problem so construction is a very important part in geometry so you have to be a little observant and always try to draw the trend where exactly construction works so that is how you will develop the snack over a period of time