 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 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? In 200 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 have 20 pens and someone gives you five more pens Okay, so now you have 25 pens Okay, so what's the increase? Increase is five. This is increase. So increase percentage Excuse 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 five Multiplied by 100. Okay, so this is five upon 20 in 200. So you got 25% increase Okay, so so if you add five 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 the 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 increased percentage or increased 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 not 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 a B C again, so hence this side is now to be 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 one So delta one, let's say the area of delta one is this and let us say delta two is area of this triangle Okay, so you can write this. Let's say delta one Is equal to area of triangle ABC Correct, which is and delta two is nothing but area of triangle a dash b dash c dash Now let's find out delta one first. So delta one is equal to under root by Heron's formula s s minus a s minus b s minus c Okay, this is delta one now for delta two. 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 the? 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 2 c will be equal to 2 small s minus 2c which is equal to 2 s 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'll 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 4 s 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 for directly so for 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 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 insight isn't it if you just double the sides you are basically if you double the sides You will get the area four times the original value. So guys. What is the increase now? increase in area It's 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 divide by original 1 was delta 1 and into 100 So if you see delta 1 delta 1 gets cancelled. So it is 300 percent Okay, so 300 percent will be Increase the area if you double the sides of a triangle