 Hi friends, so in this question it says that the you have to find out percentage increase now this is a new thing percentage increase in the area of a triangle so increase in the area of a triangle if it's each side is doubled what does this mean there are two three things which we must know here first of all let us crack what is percentage increase okay let's say you had 100 rupees okay and let's say you have 100 rupees and now you got 100 rupees from someone else so now it is doubled so you now have 200 rupees okay so what is the increase here increase we are talking about increase so increase is only about 100 rupees right you had 100 there is an increase of 100 and now you have 200 right so what is this increase on the original one is what is asked for so if you see what percentage is this increase 100 on original that was 100 into 100 that is it is 100% increase okay so if you double anything you have 100% increase okay let's take another example let's say you had 20 pence let's say you have 20 pence and someone gives you 5 more pence okay so now you have 25 pence okay so what's the increase increase is 5 this is increase so increase percentage increase percentage will be nothing but increase how much increase was there divided by on what was the original value so this was in this case it is 20 so on 20 you got an increase of 5 multiplied by 100 okay so this is 5 upon 20 into 100 that is 25% so you got 25% increase okay so so if you add 5 on to 20 so this is 25% increase now there are two things one is increase by and there is something called increased to please understand so 20 was increased by 25% to get you 25 okay now this is same thing could be said as 20 was increased to 125% why because 125% of 20 is equal to 25 okay so please be careful so increased by 25% 20 for 20 was increased by 25% is 25 and 20 was increased to 125% to so final point is 125% of 20 so which is 25 so in this case again you can find out the increase percentage or increase value was 5 okay so hence please be careful increased to an increased by okay now let's solve this problem now how to start this so let us say we'll say since there are no numerals involved so many people get confused but do not worry let's say side length is not given right so draw a triangle without fear let's say this is ABC a triangle let's say we don't know the sides no not a big deal so let's say the sides were ABC I can at least name them now they're saying the sides are doubled so A becomes 2A B becomes 2B and C becomes 2C right so a new triangle let me draw another triangle which should look at least double this size so hence it is ABC again so hence this side is now 2B let me call this as A dash B dash C dash so this is 2B 2A and 2C right now let us find out delta 1 so delta 1 let's say the area of delta 1 is this and let us say delta 2 is area of this triangle okay so you can write this let's say delta 1 is equal to area of triangle ABC correct which is and delta 2 is nothing but area of triangle A dash B dash C dash now let's find out delta 1 first so delta 1 is equal to under root by Heron's formula S S minus A S minus B S minus C okay this is delta 1 now for delta 2 let us first find out S semi perimeter for this so semi perimeter for this will be 2A plus 2B plus 2C upon 2 which is nothing but A plus B plus C isn't it now this I'm calling it as capital S so A plus B can be written as 2 small s where S is the S is whereas small s is semi perimeter of perimeter of triangle ABC okay so please remember first relation we got capital S is equal to 2 small S now for the you know the same bigger triangle A dash B dash C dash now let us find out let me do it here now S is known which is equal to 2 small s what is capital S minus 2A is it this will be S capital S minus 2A the you know for this the side length is 2A so hence capital S minus 2A which can be written as 2 small s minus 2A which can be written as 2 times small s minus A correct similarly S minus 2B and why am I doing S minus 2B because for the next bigger triangle the side length has changed to 2B hence I'm writing that and so hence it will be nothing but 2 small s minus 2B which is equal to 2 times S minus B similarly the third factor 2 C will be equal to 2 small s minus 2C which is equal to 2S minus C so hence delta 2 that is the area of triangle A dash B dash C dash is equal to capital S into capital S minus 2A into capital S minus 2B into capital S minus 2C which is equal to under root 2S into 2 times S minus A so now we will be using these these and these to replace the above form in the above formula so 2S minus B and 2 times S minus C so if you see there are four twos inside so two twos can be brought out right now because this is nothing but 2 to the power 4S S minus A S minus B S minus C which is equal to now 4 so 2 to the power 4 if you take out the square root so it will be 2 square that is so hence I'm writing 4 directly so 4 under root S times S minus A times S minus B S minus C now if you notice what is this S minus A S minus B S minus C we have just found out here this is delta 1 right the original area so hence 4 times delta 1 and what was this this was nothing but delta 2 is equal to 4 times delta 1 that means that if you double if sides are doubled this is a learning so please remember this also if sides are doubled then area becomes 4 times of original original area this is a very important inside isn't it if you double the sides you will get the area 4 times the original value so guys what is the increase now increase in area is nothing but you are saying increase in area will be nothing but delta 2 minus delta 1 the final area minus original 1 now what is delta 2 we got delta 4 delta 1 and minus delta 1 is nothing but 3 delta 1 isn't it so this is the increase this is the increase so hence what will be the increase percentage increase percentage as we discussed earlier will be the increase over the original value in 200 so in this case it is 3 delta 1 increase divided by original 1 was delta 1 and in 200 so if you see delta 1 delta 1 gets cancelled so it is 300 percent okay so 300 percent will be the increase in the area if you double the sides of a triangle