 Welcome friends, in this question of the day problem it was given that point C lies inside a given right angle and points A and B lie on its sides. So in this figure if you see this is point C which is within a right angle and O is the right angle here and A and B are on the points on the sides. You have to prove that the perimeter of triangle ABC is not less than twice the distance OC where O is the vertex so this is OC right. So basically what you need to prove you need to prove that perimeter of the triangle is nothing but AB plus BC plus CA is greater than 2 OC okay. So this is what is the question now let us see how to do it. Now in this question two underline concepts are used. One is transformation, transformation, transformation that is and which transformation we are using we are using reflection. So reflection of a point around the line and second is some of the sides of a quadrilateral is sum of sum of three sides of a three sides of a quadrilateral is greater than the fourth side fourth side this is for convex quadrilateral. Let us now understand the solution. So hence the steps are reflect point C along X and Y axis. So this is X axis this is Y axis and I am reflecting point C this is the point C here point C to along X and Y to get C dash C double dash and C dash okay and then complete the rectangle C dash C C double dash C triple dash okay. So if you see this these four points together will be will be forming the rectangle why because here is where the transformation principles are used. So if you reflect clearly C dash C is perpendicular to C C dash isn't it if you see this is 90 degrees this also is 90 degrees this is 90 degrees so hence what will happen is that from transformation philosophy we know that C dash P that is this side is equal to PC because C is being reflected so image distance will be equal to object distance similarly CQ will be equal to QC double dash right and when you join C C double dash you will get this as 90 degrees right similarly this is 90 degrees and this also is 90 degrees. So basically C C dash and C C double dash are perpendicular to each other. So when they are perpendicular to each other I can complete the rectangle so hence this angle is also 90 degrees okay now join OC dash and OC double dash and OC triple dash all from point O our joint now if you consider these two triangles I am going to highlight the triangles CP so this triangle these two triangles if you take this is one and this is the one of the one okay if you see they are congruent triangles why because PC dash PC this this side is equal to this side and this is 90 degree and this side is common OP is common so hence they are congruent that means what all inferences will draw so OC dash this side this side will be equal to this side first of all that and then theta one will be equal to theta two isn't it similarly if you take these two triangles which two OQ QC dash and OQ C triple dash if you take these two triangles again in that also you can prove these two triangles to be congruent and three to three will be equal to theta four and OC will be equal to OC double dash what I'm saying is OC will be equal to OC double dash right so now if that is so so you can see from this we can say that theta one and is equal to theta two and theta three is equal to theta four but theta two plus theta three was 90 degree isn't it theta two plus theta three is 90 why because it was given this was point O was a 90 degree and now theta two is equal to theta one so I can replace theta two by theta one and theta three by theta four why because it's given here and here so hence theta one plus theta four is also 90 degrees that means all angle put together theta one plus theta two plus theta three plus theta four is 180 degrees that means this line is straight line C dash O C double dash C dash O C double dash is a straight line right now so and O is also the midpoint clearly if C this is you know because OC dash is equal to OC is equal to OC double dash so O is the midpoint of C dash C double dash now C dash C double dash happens to be the diagonal of rectangle a given rectangle right C dash C C double dash C triple dash so of this rectangle which rectangle if I take this rectangle this whole this one so C dash C triple dash is a diagonal similarly O is the midpoint and you know if you join OC triple dash then O C triple dash and C all will be collinear only why because O is the midpoint and the diagonals of a rectangle bisect each other right that means if O O will lie on both the diagonals right O will lie on both the diagonals so O is already on C C dash sorry C dash C double dash so O will be lying on the other diagonal as well so this is also the diagonal where O is the midpoint okay now so yeah so now in quadrilateral C dash ABC which quadrilateral I am now taking I am taking this quadrilateral this one this whole this whole this quadrilateral I am taking right in this quadrilateral the three sides that is which three sides I am taking let us say this yeah so this these three sides C dash A then C dash A then AB then BC double dash and then finally C double dash C dash okay I will draw again C double dash C dash in this triangle the quadrilateral C dash A so point A is here so C dash C dash A plus AB plus BC double dash is greater than C dash C double dash that's what we have written here just one now C dash A is clearly AC if you see C dash A A is equal to AC we are just two triangles to the quadrilateral so C anyway C is the reflection of C so AC dash will be equal to AC similarly BC this BC will be equal to BC double dash isn't it so that means I can replace C dash A by AC let AB be AB as it is and BC dash can be replaced by BC adding all together we will get this is more than the fourth side that is C dash C double dash so that's what I have written now in a rectangle I can replace one diagonal by the other diagonal so the diagonal is C double dash C triple dash C correct and C triple dash C if you see the figure C triple dash C in the figure is nothing but twice of OC why because O happens to be the midpoint of the two bisecting diagonal so hence I can replace C dash C triple dash C by 2 AC and hence the result right the parameters of the triangle is better than 2 AC