Hi everyone, my name is Ravi Prakash and this is the 10th class of Geometry. Okay, so here we'll see finding number of triangles when perimeter is given, right? When perimeter is given, how to find number of triangles, right?
See. I'll tell you a super shortcut here. If perimeter is given, right? So if perimeter is given, so perimeter, if it is odd, and when if it is even, right? If it is even and when it is odd, right?
When it is even, let perimeter be P, let perimeter be P. So I can form P is. square by 48 distinct triangles, distinct triangles. And when perimeter is odd, we can form P plus three square by 48 distinct triangles. Now, these things come from higher math actually right from higher math this formula comes right. So, but we don't need derivation, right?
There's a person called Craig Byraman, if I remember. Okay. He derived this formula and it's a pretty good formula.
I can directly use in updated examinations because in previous year CAT also, in ZAT also, in all your mock test also, right? Many times this kind of questions are asked, right? That if some perimeter is given, let's say perimeter is 20, how many triangles can you form?
So, by putting those conditions that sum of two sides is greater than third side and all, we can do it. No issue. But no use of doing it, right?
Because when we have a... Direct formula to do it, right? We can simply proceed it.
Proceed this. Okay. So, I'll tell you, now this bracket is what? This bracket.
This bracket is a nearest integer function. Okay. This bracket here is very important, right?
This thing. This bracket is a nearest integer function. Right.
Nearest integer function. Now listen carefully. A nearest integer function that basically means what? If the bracket value is 3.8, our answer will be 4. Nearest integer.
Okay. If the bracket inside value is 3.1, our answer will be 3. Got it? If bracket inside value is 3.4, our answer is still 3. Because nearest integer to 3.4 is 3. If bracket inside is 3.58, our answer will be 4. Because no? Greater than 3.5?
the nearest integer will be what? 4. Right? Now, the conflict will come when bracket inside bracket when it is 3.5 exactly, right?
So, when bracket inside is 3.5, the value will be 3 only. Right? So, in this case, the value is 3 only, right?
So, if bracket inside is 3.5, the value is 3 only. Okay. So, this is what?
This is nearest integer function. So, finding number of triangles when perimeter is given, The first case is when perimeter is p. So, if perimeter is odd here, so I can form p plus 3 whole square by 48 distinct triangles inside this nearest integer function and p square by 48 distinct triangles inside this nearest integer function, right? Now, suppose the question here is, the question is, if perimeter is equal to 16, how many? how many distinct triangles can be formed?
How many distinct triangles can be formed? Right now, very easy. So perimeter is even here. So when perimeter is even, I can directly apply that funda p square by 48, right nearest integer function should become 16 square by 48. Okay, that is 256 by 48. Now we just have to check that is value is less than something point five or more than something point five, right? Because if it is suppose it is 48 here, I know that 48 into five is 240. So if it is 240, if it is five, it is obviously five point something right and 48 into six is 288. Here it is 256 it is five point something right.
So, if it is greater than 5.5, answer will be 6. And if it is less than or equal to 5.5, answer will be what? 5 only, right? We just need to check it is less than 5.5 or greater than 5.5, right? So, obviously, it is less than 5.5 because 48 into 5 is 240 and 48 into 0.5.
That is what is half of 48. Half of 48 is 24. So, 264, right? 264. So, this will be 0.5. So, 48, sorry, 5.5, 48 into 5.5 will be?
48 into 5, that is 240, plus 48 into 0.5, that is half of 48, that is 24. That is 264, is what? 48 into 5.5. So, it is 256, it is obviously less than 5.5.
So, we can write, okay, roughly write around 5.4. So, it is 5.4, what is the answer? Answer will be 5 only. Because it is the nearest integer function. Right, nearest integer function.
Okay, so answer is 5. Answer is 5. Similarly, another question. If P is equal to 27, okay, so how many distinct triangles can be formed? Okay, so in this case now, P is odd here, right?
P is odd. I can directly use that formula. P plus 3 whole square by 48 inside the nearest integer function.
Right? What is PR? 27 plus 3. That is 30 square by 48. Now what is 900 by 48?
So 900 by 48, you see roughly it is how much? It is roughly 8 point something. Right?
To be exact, it is 8, roughly it is 8 point something. Right? Yeah, 8 point something. 960. So but 8, it is, is it more than 8.5 or it is less than 8.5?
Right? So you see, No, not sorry, not 8.5. I should say it is 18.5, right?
So it is more than 18.5 or less than 18.5. Just have to check, right? What is 48 into 18? So you see 48 into 18 is nothing but can directly do, sorry, 48 into 18. Do it mentally.
48 into 18 is, see 18 into 50 is 900 minus 18 into 36. it is 864, right? And half of 48 is 24. So, 864 plus 24, 888, right? So, 18.5 is 888. It is 900 here.
It is 900, right? So, it is more than 18.5. So, let's say it is around 18.6. So, what is the nearest integer function?
I hope you got it, right? So, very easy to find it. Okay.
Next case, next case will be for a scalene triangles. Okay, it will be for a scalene triangles. For a scalene triangles.
Same funda, how many scalene triangles I can form, right? So again same thing, if perimeter is P, if perimeter is P, or you can directly write for scalene triangles, simply, simply, Replace P with P-6 in the previous formula. Previous formula, right?
Notice the Scalian triangle. Scalian triangle means all three sides are distinct. Right? All three sides are different.
So, if perimeter is P, so what that formula will come here? That formula will come now if perimeter is P. So, when P is odd, Right. So it was in that in the previous case, it was what? P plus 3 whole square.
If you replace P with P minus 6, it becomes P minus 3 whole square by 48 inside a nearest integer function. So when P is odd, I can form P minus P minus 3 square by 48 is scalene triangles I can form. Scalene triangles, right?
And when if P is even, if P is even, so again in the even. For earlier in total triangles, what is the formula? p square by 48. So if you replace p with p minus 6, if you replace p with p minus 6, you get what?
p minus 6 whole square by 48. Scallion triangles I can form, right? This is the, this is the funda. For odd p minus 3, for even p minus 6. Okay. So how many? Scallion triangles, you can do a question here.
How many? A scalene triangles can be formed. How many scalene triangles can be formed if, if question one, that is A part, if perimeter of the triangle, if perimeter of the triangle is equal to, let's say 24. So if perimeter of triangle is 24, how many scalene triangles can be formed, right? Now P is even, right?
Here, p is even. So if p is even, simply apply the formula p minus 6 whole square by 48 inside the nearest integer function for p is even, right? That is how much? That is 18 square by 48. 24 minus 6, 18 square by 48, right? Notice 324. That is 324 by 48, right?
324 by 48. How much it is? So again, 48 into 6 is 288. That is more than 6.5. So, let's say around 6.6.
That means 7. 7 is the answer, right? So, if perimeter is 24, I can form 7 scalene triangles. I can form. Okay. Same way, let's say if perimeter is, if perimeter is like 17. If perimeter is 17. Then how many scalene triangles I can form?
Again, same funda. In this case, perimeter is odd. So, if perimeter is odd. what I can do simple apply the formula p minus 3 whole square by 48 right that is 17 minus 3 14 square by 48 right that is 196 by 48 so 48 into 4 is 192 just around 4.1 right what is the answer answer will be 4 so answer will be 4 for this question right so this we can directly find If perimeter is given, I can easily find total number of triangles.
I can form total number of scalene triangles as well, right? This will help me if I have to calculate isosceles triangle as well as equilateral triangle. This will really help me, right? Directly, I can extend this concept.
Okay, so how many isosceles? I can write this next question. How many?
It's a very good concept, right? Directly, you can find the answer. How many? isosceles triangles, how many isosceles triangles can be formed, can be formed if, if perimeter is given as 30. If perimeter is given as 30, how many isosceles triangles can be formed, right?
See for isosceles triangle, we don't have direct formula, right? But I can directly, I can easily do that. I know that Total number of scalene triangles plus isosceles triangles, right?
Plus equilateral triangle. What are scalene triangle? All three sides different, right?
What are isosceles triangle? Two sides different. Sorry, two sides same. And what are equilateral triangle? All three sides same, right?
So, scalene triangle plus isosceles triangle plus... equilateral triangle number of obviously number of number of the scalene triangles plus number of isosceles triangle plus number of equilateral triangles is equal to total number of triangles obviously right so for getting number of for getting number of isosceles triangle the funda will be for getting number of isosceles triangle funda will be from total number of triangles from total number of triangles Subtract, subtract number of scalene triangles plus number of equilateral triangles. Okay, that's it. So, I'll get the number of isosceles triangle. I know how to get number, total number of triangles.
I know how to get number of scalene triangles and number of equilateral triangle is very easy because all three sides are different, right? So, if perimeter is 30, there will be only one equilateral triangle besides 10, 10, 10, that is 30 by 3, right? 30 by 3 is 10. So, 10, 10, 10, only one equilateral triangle, right?
So, first, we should calculate number of isosceles triangles. So, first, we should calculate total number of triangles, right? So, how many total number of triangles?
So, total number of triangles, total number of triangles. Now, here P is even. So, if P is even, it is how much? It is P square by 48. Inside nearest integer function That is 30 square by 48. Now, it is how much?
We just discussed it. It is around 8 point, it is more than 8.5, right? So, if it is more than 8.5, let's say around 8.6, the value will be what? 9. The value, sorry, sorry. Why I'm doing 8.6?
18.6, sorry. 18.6. Okay, what is the value?
It will be 19. So total number of triangles is 19 with perimeter 30. Fine. Okay. Number of scallion triangles. How many number of scallion triangles I can form?
So number of scallion triangles just replace P with P minus 6. It is P minus 6 whole square by 48 inside the nearest integer function. Okay. So this is 30 minus 6, 24 square by 48. So, it is 4c, it is 470, sorry, 576. Okay.
So, 24 square is 576. So, 576 by 48 is exactly 12, I think. Right. It is exactly 12. Right.
So, it is exactly 12. 24 square by 48 is exactly 12. So, what is the answer? Can you tell me quickly? 12 only. Because no decimal part. Right.
So, if 12 is the answer, so 12 is only. So, total number of square and triangle is what? Is 12. Okay.
Yeah, fine. Now, number of equilateral triangle. Number of equilateral triangles, right? Only one. Only one.
Why? Because it is perimeter is given as 30 here. So if total perimeter is 30, I can form only one triangle with sides 10, 10 and 10, right? That is 30 by 3. Okay. Suppose perimeter was 20. In that case, no equilateral triangle with integral sides.
No equilateral triangle with integral sides, right? So, I cannot form any integer side. These all are obviously for integral side, right? So, these are all for integral side. So, how many?
It's a scalene triangle side. See, otherwise there is no question, right? How many triangles? The question is, if perimeter is 30, how many scalene triangles can be formed? Infinity, right?
Because there is no point in decimal going in decimal. 24.1, 24.2, 6.1, 6.11, 6.21, right? So, decimal's question will not be valued one, right? So, it has to be for integral sides. Okay.
So, if side is 20, sorry, if perimeter is 20, there will be no equilateral triangle with integral side. Because 20 by 3 is 6.66. Okay. So, we will only get equilateral triangle with integer sides if perimeter is a multiple of 3. Okay.
If perimeter is a multiple of 3. So, number of equilateral triangles is how much? 1. So, what is the answer here? We simply place the values.
Number of isosceles triangles is equal to total number of triangles. How much we got? We got 19. This is 19. Okay. So, I can write here.
We write here 19 minus number of isosceles triangles. That is 12 plus number of equilateral. That is 1. So, 19 minus 13. What is the answer?
Answer is 6. Answer is 6. Right. So, very good question. Right. An application of this funda that when perimeter is given, how to find number of sites and all.
Right. In any triangle. How to find number of sites.
Okay. Bye. Now, let me discuss more questions here. Now, just write next question.
How many, how many distinct triangles can be formed with integral sites? How many distinct triangles can be formed with integral sides if p is equal to 80? Okay. Now, we'll put the condition as sides as being even and odd, right?
So how many distinct triangles can be formed with integral sides with integral sides, okay, if perimeter equal to 80 with question one with sides even with sides even and question two, two sides odd, two sides odd and one side even. Now it's a good question. Right? How many distinct triangles can be formed with integral sides if p equal to 80?
Right now just pause this video. and try it for 2 minutes, right? Try it for 2 minutes.
It's a good question. Now we'll solve it, right? See, so how many distinct triangles can be formed with integral sites if P equal to 80 with sites even?
So you can see how to write it, right? See, we should write like this. Suppose sites of a triangle are A, B and C.
I should write, okay, A plus B plus C is equal to 80, right? But it is given that sites are even, that means A is A, B and C are even, right? So, I should replace A with, how to make A even? How to make A even?
I should replace A with 2A. I should replace B with 2B. I should replace C with 2C.
This is equal to 80. Now I made it even, right? That means A plus B plus C is equal to what? It is 40. Okay, so now how many triangles can be formed with capital A plus capital B plus capital C equal to 40, right? Now see what is capital A? What is capital A right now?
Capital A is obviously when A is, when we made A even, so now it will only take the even values of A, right? So, A plus B plus C equal to 40. How many triangles I can form? So, total, since side is even, so total triangles I can form is, I can form is P square by 48, okay?
That is 40 square by 48. That is 1600 by 48. Now, what is the nearest integer function? 15 divided by 48, right? What to do now?
So again, sense it right. It is close to which number? You can simply do 48 into 3 is 144. 16 and 0, 60 is left. Okay.
So again, 48 into 3 is 144. And again, 16 is left. So now 16 is less than half of 48, that is 24. That means there's something less than 0.5, 33.4 something I can write. Right.
Simply divide it. 48 into 3 is 144. So 16 is left. Now 16 and one more zero again 160. So again, 48 into 3 is 144. So again into three, now 16 is left right now obviously 16 by 48 is less than 0.5 right? Is it exactly 0.33 but no you check it at 0.4.
What is the answer? Answer is 33. So answer is 33 for this question. Right?
So answer will be 33. Yeah, I hope it is fine. Right now, two sides or second question is the answer of the first question, right? Now, second question is two sides odd and one side even, right? And let me wrap this part now.
Second question, two sides. I think the thing on it, right? Two sides odd and one side even.
What does it try to represent? Right? Two sides odd and one side even. See now, second solution I'm writing solution to see, I know that again, A plus B plus C is equal to 80, right? is equal to 80. So if a plus b plus c is equal to 80, so this c, this total is always even, right?
So when summation of three things is giving me an even number, listen carefully, right? Very point concept here. When summation of three things is giving me an even number, so it has to be what?
Either the first case, right? What is the first case? The first case could be all are even, even and even.
So, even plus even plus even will give me even. Right? First case. What is second case? Now, all three can't be odd.
Right? All three can't be odd. Right?
But, but, if one side is even, so other two can be odd. Odd plus odd plus even will also give me even. Odd plus odd plus even will also give me even as 80 is an even number. Right?
So, odd plus odd plus even will give me an even number. Because odd plus odd is what? 3 plus 5 is 8. 3 plus 5 is 8. So, odd plus odd is even.
So, odd plus odd is even and even plus even is not even. Right? That basically means what?
Only two cases. Understand, only two cases. Right?
When A plus B plus C is 80, there could be only two cases. In first case, either all are even. In the second case, two sides are odd and third is even. See, think of any other case, you won't get an even number. If you make A as odd and B and C is even even, So, odd plus even is odd and again odd plus even is odd, right?
So, in any case you'll get odd number, you won't get 80 here. You won't get 80 here. This is an even number, so only two cases, right?
So, total answer, what I should do? I should calculate the answer for total. What is P square by 48, right? That is 80 square by 48. Fine. What is the answer for this case?
Answer for this case is what? When all are even, even, even. We discussed it. What is the answer for this case?
It was 40 square by 48. We discussed, right? What was the answer? 33 was the answer, right? So, 40 square by 48. is the answer for this question, right?
So, out of total triangles 80 square by 48, 40 square by 48 are for case 1. So, that means remaining will be for case 2. Remaining will be for case 2. What is remaining? What is remaining? So, remaining for case 2. That means 2 sides odd, right? 2 sides odd and 1 even.
and one even, right? How much it is? 80 square by 48 total number of triangles minus 40 square by 48. Right, 40 square by 48, right? Varying point concept because this A plus B plus 80 comprises only two cases, right?
It consists of only two cases. What are two cases? Since result is even, So, only even plus even plus even or odd plus odd plus even, right?
So, when this 80, what is the total angle of triangle? 80 square by 48. Then, simply solve it, right? So, just solve it.
A little more easier. 80 square by 48, right? What's the value here? So, value will be how much? It is like 80 square.
So, 6400 by 48, right? And simply solve it. Again.
48 times 1 is 48. 16 times 3 is 144. Again 60, so 133. Right? So, 133.3 is coming around. That means it is around 133 triangles.
Around 133 triangles here. And here, how much is the triangle? 33 triangles. Right? 33 triangles.
Right. So, what is the answer? The answer is what?
The answer is 133 minus 33. The answer is what? 100. So, answer for the second question is 100. Answer for this first question is how much? 33. Total number you're trying is how much? 100 plus 33, 133. Right?
Only 2 cases. Think of it, think of it. Only 2 cases. Either all 3 sides are even. Or second case, Two sides are odd and one side is even.
So when 80 square by 48, that means total there are 133 triangles can be formed. Out of 133 triangles, 33 triangles is in the first case. That means remaining 100 should be in the second case.
That is odd plus odd plus even. Okay. Okay.
Thank you. We'll continue this in next video. Okay.