 Hello friends, so welcome to this question of the day problem Now it says there's a figure given and it says that a person wishes to move from point P to point T via a point R on line QS. This is the line QS if you see and Yeah, this is QS and Person starts from P and he has to go to P right via point R such that And it's given that the lines PQ and TS are perpendicular. So and RT are straight lines, right? All these are straight lines. We are given that PQ is 5 so you can see 5 and TS is 3 and QS is 10, okay And you have to find out the distance QR such that the total distance traveled that is PR plus RT is minimum Interesting problem. So in this problem, we are going to use transformation and or you can also say we can we are we are going to use principles of Physics as well. So and that is We will use the concept of reflection, right? So how to solve this problem? So imagine that this is the line QS we are talking about and here is Here is my point P. Now many of you have thought that point R is exactly the midpoint of Q and you have taken midpoint R as midpoint and join them and then you have find out PR and RT using Pythagoras theorem But that will not be the right solution. This would be R would be midpoint only when PQ and TS that is this length and this length would have been same Then R could have been the midpoint, but since they are different R will not be the midpoint. Now, how to go about it then? So We are we are we are given to find out We are given to find out The distance PR plus RT. So I'm going to write it here PR plus RT to be minimum to be Minimum now the moment this word minimum comes It strikes my mind that Minimum is what is the minimum distance between two two points? It's a straight line, isn't it? But in this case it's not a straight line. So what to do? So this is the technique. So what you can do is imagine this to be a mirror and reflect point T on This mirror. So you will get point T dash here T dash here, isn't it and you join So if you see PR RT dash will be a straight line, right? So PR and RT dash, sorry PR and T dash are collinear PR and T dash points are Collinear. So what we have done is we have reflected point T All on QS to get point T dash, right? So now that now, you know when it is reflection case of reflection then T s will be equal to T dash s Isn't it? Also, we know that these are perpendicular So hence this angle is equal to this angle as 90 degree and RS is common. So hence I'm going to write this that in triangle PRS and Triangle T dash RS Angle S is equal to angle S both equals to 90 degrees correct PS is equal to ST dash and this is by construction. I have reflected and Deep I've reflected point T to get T dash and RS is equal to RS, which is common side Common side, isn't it therefore by SAS criteria, what can I say? I can say triangle TRS is congruent to triangle T dash RS The moment it is congruent. So now I can say this angle is theta So this angle will also be theta because they are corresponding parts of congruent triangle. So angle TRS Is equal to angle T dash RS is equal to theta Isn't it? And also RT is equal to RT dash Now since PT dash is a straight line that means That means PT dash is the smallest distance between P and T dash right so PT dash is smallest in length in length or PR plus RT dash is Smallest in length Why am I saying that because PT dash if you see is nothing but PR plus RT dash. This is RT dash and This is PR. So PR plus RT dash is PT dash, which is the least now from here If you see RT is equal to RT dash. So hence I can replace this as PR plus RT is smallest or the minimum or the minimum Hence that is what is required. So hence from what point to what point it should be So hence where should we point R? Point R should be nothing but reflect T to T dash and join PT dash. So wherever it is that is R now we will also use the concept of similar triangle to now find out X how so if you see triangle PQR is similar to triangle T dash SR why Because we have these two angles as same that is angle PQR is equal to angle P Sorry T dash SR which is equal to 90 degree both and angle PRS Sorry PRQ is equal to angle T dash RS. They are vertically opposite angles, right? So by AA criteria Angle angle criteria, we know that These two triangles are similar. So hence we can say PQ by so I'm writing here now for simplicity. So here you can write now Yes, so PQ upon ST dash is equal to QR upon RS why because corresponding sites now will be proportional So PQ by ST dash so you can write 5 upon 3 is equal to QR QR is 10 minus X upon X Okay, so you can simplify 5x is equal to 30 minus 3x so 8x 8x is equal to 30. So X is 8 30 by 30 upon 8 which is nothing but 3.7 5 Right. So hence X should be at point R is R is 3.75 units away from S on Qs Right, so this is the solution So what did we learn? We learned that the smallest possible distance between two points is the straight line So hence you have to just apply this trick. You have to reflect reflect T dash T on to T dash T on to T dash and then connect P T dash and you can then solve Using similar triangles. Thank you