yes I confirmed, it would have being the fastest method.

Yes sir it's always advisable to bring roots at the top, to make solving easier thats why when ever you have a root as a denominator you just multiply the whole fraction by another fraction, with both the denominator and the numerator equal to the root in the original fraction, let's say if you multiply by 1/√a by √a/√a it will be √a/a But multiplying by √a/√a will not change the value because √a/√a = 1 So it is as good as multiplying by 1 I hope the explanation is clear but Please if it isn't clear enough, just tell me, I'll reexplain, than you sir

In American classrooms we call this step rationalizing the denominator. Sometimes a clever choice is required to accomplish that such as using the complement of an irrational sum to give a difference of squares.

Ok sir, the small triangle has its angles to be 90°, x and 90°-x The big triangle also has its angles to be 90°, x and 90° - x. That proves that the triangles are similar, and if they are similar their sides must be in the same ratio. That's why in the small triangle the side facing angle x which I labeled "b" divided the side facing angle 90°-x which I labeled "c" will equal the side of the big triangle facing angle x which has the length 10m divided by the side facing angle 90°-x which has the length (10m+10m=20m) That's why b/c=10m/20m. Another way to confirm it, is by using Tan∅ = opp/adj If you use it on the small triangle Tan(x) = b/c And for the big triangle Tan(x) = 10m/20m So therefore tan(x) = b/c = 10m/20m I hope my explanation is clear sir, but if it is not clear enough please tell me, I'll reexplain it. Thank you sir

Thank you for this Explanations good job ❤️

Thank you sir, it's my pleasure

Ok I'm trying to understand what said but I am not sure I understood you correctly, you said you used or formula, and you got 783cm, please which of the length, did you get as 783cm Or are you trying to say that, you used the final result to calculate the area of the big square and you got 783cm²

@@MathandEngineering THE AREA on 784cm... I did an improvised calcul (formule I developed now) and the solution was 783cm...

I TOOK THE 81cm2, and I started to do the calculs. And I had 783cm as solution...

And there are no digits after the decimal point, you got a perfect 783cm²

@@MathandEngineering A speed calcul... My formule runs that way. To have an aproximation and some basis near to a speed solution.

Glad you like it!

Ok sir, please is there any part you want me to reexplain?

Well sir, it also took me Quite some time solving it. I discovered upto 5 different method to solve it, but I just had to pick 1 or 2 method, to present,

I am sorry I even forgot to tell you that Question I asked for your opinion the other day, just like I said it was impossible, After some conversation with the person who asked the Questions, we realized that he had a slight typing error which made it a different Question to me the reader, but I got the other Question and he even proved it himself. So the Question I wrote was incorrect

@@MathandEngineering I think you must have much more time than I 🙂. Instruction, grading and planning occupy about 75% of my waking time.

Yes sir, I am kind of less busy, I don't have any much commitment right now, except for solving math, so I am using most of my time in solving math

Yes this is true, even I realized that when I got to that point, that somebody may confuse the x of the Quadratic formula as equal to the value of angle x, I wished I had noticed before I started making the video

Elementary

Wow the method is very good and also creative, I could have solved it using this method, I'll make sure I try this approach of yours on similar Questions, I am sure it will produce a precise answer like this

@@MathandEngineering thanks, bring problems from Jee advence, please

Ok I'll go look for the Question, and please, if there is any Questions you want me to solve, you can also put it in the comment section, I'll solve it, thanks

​@@MathandEngineeringIn triangle ABC, show that sin( A /2)≤ a/b+c Try this .

Well yes sir, the Question was kind of tricky, when I saw the Question, I also had to play with it for sometimes before I miraculously arrived at an answer 😂

Good day sir, well you can use the formula ½b×h also, but to us that, you have to choose one of the sides as your base, than calculate the height, that is a line perpendicular to the base which touches the vertex opposite to the base, though in the case of a Right angled triangle, the two perpendicular sides, each can be taken as the base or the height. But the triangle from the Question is not a Right angled triangle, because we calculated angle y to be equal to 180° - x, angle x is also one of the angles of the other right triangle, in a Right angled triangle, apart of the right angle, none of the two other angles can be up to 90°, the two are less than 90°. They are complementary. So this shows that x is less than 90° and we found y = 180° - x if you subtract x which is less than 90° from 180°, definitely the answer will be greatery than 90°, there fore y is greater than 90° and hence we have to use the formula ½absinC. I don't know if my explanation is clear,.but please if maybe you didn't understand, just tell me and I'll re explain. Thanks

Yes sir, those two Right triangles that formed the kite, are Congruent, because two of All their angles correspond and they have 1 common sides which was the side I called the diagonal of the kite.

Okay so you mean, you solved it without even a pen and a paper, like you did all the solving in you mind, wow that's really amazing. As it is, experience is the best teacher

@@MathandEngineering yes! I try to emphasize mental math in my classroom largely because American students are so very technology dependent, with many lacking much number sense at all. I have been using Prof. Arthur Benjamin’s methods, detailed in his book “No Calculator? No Problem! Mastering Mental Math”

Good day sa, i wanna ask for your opinion, a friend asked me to prove from a triangle ABC that sin(A/2) ≤ a/b + c Which I understand to be sin(A/2) = a/b + c. Or Sin(A/2) < a/b + c So I found it impossible that sin(A/2) = a/b + c Because the sine ratio of any angle can never be equal or greater than 1, it is always less than 1, But in a triangle A B C, no matter what it's length are, a/b + c is always going to be greater than 1. I told them that it was impossible but he insisted that it was possible, That's why I am asking for you opinion, am I missing something?

@@MathandEngineering I need to take some time working on this before I answer.

Ok thanks sir, I'll be waiting for your answer,

@@MathandEngineering I did not know how to compute a root until I was studying engineering. I would use a calculator or a log table.

You know using calculators and These stuffs, makes math boring. Math is more interesting when you are solving it using knowledge rather than using a calculator. I don't mean using a calculator is bad, but I noticed that if you solve if yourself, it makes you more interested in learning more things and even curious about everything in the world, and that will make you acquire alot of knowledge. Just like Sir Isaac Newton who discovered gravity, because he was curious about why the Apple fell. That is the same for math lovers I noticed we all are curious as to why is this so. When something happen you don't ask yourself, but your heart does. This really helped me alot. Being curious made me know many things that I wouldn't have known if I wasn't curious

@@MathandEngineering very true! I recognized this fairly early on in my own life, but teachers were annoyed to be drawn off topic. As a teacher I oppose the use of calculators until the necessary computation is well understood and ideally some skill has been developed. (Sadly this seems impossible computing trig values.) For example, when my students saw that sqrt(0.25)=0.5, I asked them if that seems right. They couldn’t say yes or no because they had no sense of the function. I next showed then the square roots of 25, 16 and 4. ONLY THEN did they (wrongly) conclude square roots are smaller. After understanding the root of 1 is 1, they understood it’s not so simple. Then the graph y=sqrt(x) made sense to them and they learned a great deal that day. Calculators undermine learning if they are not used judiciously.

Yes the graph is really going to make sense and convincing to every one. You know when I am dealing with numbers behind the point, I like writing them in standard form, for example to calculate the square root of 0.25 I'll just write it as 25 × 10^-2 It's square root will be √(25 × 10^-2) = 5×10^-1 = 5×0.1 = 0.5

Yes sir, the moment the line is drawn, the Question Becomes solved completely, there once has being a particular in my phone for more than a year, I thought I was going to use trigonometry to solve it, and I didn't want to ask anyone, I kept trying, and trying, untill i finally gave up, and ssked someone. I was shocked that the moment I gave the Question, in some seconds he has already another line, and the moment he drew it, even without him completely solving it, I saw the answer.

I have a problem for you. I’m looking for an elegant solution. I know the answer but it took me far longer than I’d like to admit: Two women set out at sunrise and each walked with a constant speed. One went from A to B, and the other went from B to A. They met at noon, and continuing without a stop, they arrived respectively at B at 4 pm and at A at 9 pm. At what time was sunrise on that day? Let me know if you want the answer. I’m hoping you will solve it beautifully and perhaps make a video of it.

After calculating I found sunrise to be at exactly 6am, am I correct?

@@MathandEngineering Absolutely correct! Did you end up using 5 equations?

Actually the way I solved it, I named the point where the women met at noon to be C, now we have two distances, A -> C and B -> C. Since each of the women was moving at a constant speed, I assumed the speed of the first woman moving towards B to be P and that of the woman moving towards A to be Q. Now we know (distance = speed × time) The first woman moved from A to C at time "x". So AC = P×x = px The second woman moved from C to A at time (12:00am to 9:00pm) = 9hrs so AC = Q × 9 = 9Q Therefore Px = 9Q The first woman moved from C to B at Time (12:00am to 4:00pm) = 4hrs BC = P × 4 = 4P While the second woman moved from B to C at time (sunrise to noon) = x BC = Q × x = Qx So 4P = Qx (Px = 9Q) ÷ (4P = Qx) x/4 = 9/x x² = 36 x = 6hrs 12:00am - 6hrs = 6:00am = sunrise

Yes you got it, because if you calculate the value of a, it's 8m thereby making the triangles which formed the parallelogram to be equilateral. How did you figure out that the two triangles are equilateral without calculating, I am really impressed. All the time I have to use theorems and rules to know the shapes

@@MathandEngineering they just looked equilatral-it was a guess. I draw them often for my geometry classes, but I can’t tell whether they are or are not when they’re off by less than 2-3° (such as a 59-59-62 triangle).

Ok do you mind if I prove it?

@@MathandEngineering Yes please, want to see the proof.

@@mohabatkhanmalak1161 ok first draw a Right angled triangle ABC A B C Assume the length of AB to be x and BC y Using Pythagorean theorem AC=hyp=√(x²+y²) If you draw the median from the right angle to the Hyp, AC will be bisected, if we assume the point where the median touches the hypotenuse to be P AP = CP = ½√(x²+y²) now a draw line parallel to AB from point P down to BC and label the point Q. We know that any line that touches 2 sides of a triangle and is parallel to the other side will divide them in the same ratio, so since PQ bisects AC it will also bisect BC, therefore BQ=CQ=½y Now we know also, if a line bisects two sides of a triangle, it's length will be half the length of the other sides so PQ = ½x Now look at triangle BPQ P B Q BP = hyp(BPQ) = median of ABC BQ = ½y PQ = ½x Using the Pythagorean theorem (BP)² = (½x)² + (½y)² = ¼x² + ¼y² = ¼(x² + y²) = (½×½)(x² + y²) √(BP)² = √((½×½)(x² + y²)) BP = ½(x² + y²) Recall BP is the median of ABC AND AC is the hypotenuse of ABC and it's length is √(x² + y²) BP is the median from the right angle, and it's length is ½√(x² + y²) Which is half that of AC this proves the Theorem that the median of a Right angled triangle from the vertex of the right angle is half the length of the hypotenuse I hope you understand the explanation, if you don't. Then please tell me, I'll make a video to make it clearer, thank

I wish KZbin allowed us to respond with graphics and videos. As I was reading your proof I found myself thinking it seems circular and I am not sure I “know” some of your assumptions. I watch a lot of KZbin math content, and their policy to prevent us from sharing disrupts effective intellectual dialogue. Too bad there’s not an open math forum where we can pose questions and dig our own rabbit holes. I know of some things that come close, but I hesitate to even mention them lest I get blocked.

No I guess, you were only tired, maybe You've been working alot, and you got tired. You know sometimes even I, after uploading a video, I just find myself closing my eyes even though I am not feeling sleepy, and I say "ohh, the eyes are tired and the body is still active"

Please sir is frequen consumption of energy drink, dangerous to the health. From my teenage I have being addicted to drinking carbonated drinks, when I kept hearing from people that it's unhealthy, I attempted stopping but unfortunately I got addicted to energy drinks and now I am worried, any day I attempt to avoid energy drink, I'll just be down and won't be able to complete any task from the tasks I have.

Thanks

Good day sir, thank you so much sir, I am really grateful, you saved me today, I can't imagine how many people would have seeing the video and get the wrong information, I wouldn't have realised if you hadn't told me, because after uploading it, i wasn't coming back to watch I would be busy solving more math Question, thank you for the correction, the Approach you posged there was the best, I didn't want to take down the videos with out your permission because your comment was there, i just had to take it down immediately, because if I left it there, some people who are watching to learn may watch and not notice the mistake like you did. Though I have screenshoted the method comment, thank you sir I appreciate. I think I solved the Question in a hurry, because I can't imagine how I solved it made the video and uploaded it without noticing anything wrong, it was a mistake forgive me for that sir

@@MathandEngineering I’m honored-you’re a gentleman and a scholar!

Thank you sir, I'll be very happy If I become a scholar

@@MathandEngineering As a teacher let me assure you, this is something you’ve already achieved. If anyone challenges themselves to learn more and harder things, that’s the core of true scholarship. On the other hand, gaining the recognition of academic institutions is a wholly different matter. There are many very wealthy people constantly earning upper-level degrees more by virtue of their wealth (and the vast leisure time required) than as a result of any ability and/or diligence on their part. If a Ph.D. is your goal, work toward it, but if discovering mathematics is the joy of your life, you might not enjoy having “experts” charging you a premium to interrupt your accomplishments by pressing you into courses that bore you and research that primarily benefits them (certainly more than you).

Wow thanks sir/ma 🤗

Ok sir, I have actually tried another method, but it turned out to be as long as the one in the video, yes d = 2a, but I don't understand which of the triangles you were referring to, when you said the large triangle is isosceles, are you referring to CST because it's the only isosceles triangle there, none of the other triangles is isosceles. Ok I'll still try solving using other methods, to see if I can get a shorter solution. Thanks

I may have made a mistake, I had erroneously reasoned triangle ABC was isosceles but can’t reproduce it. I’ve now tried it again. Hopefully the right triangle’s legs are 14 and 8sqrt(2) with hyp=18.

😂, angle "a" kept waiting for me, but I shocked it That's the true definition of "never let them know your next step"

Wow thanks 👍

Thanks

Thank you so much sir

i did it in completely different style

I also solved the problem in my head.

Good day sir, pls why do you think that we can't use small letters to lable point, there are no differences between small letters and capital letters in terms context. They are just us to name points.

​@@MathandEngineeringsmall letters are to denote lengths and capital letters for points.

Ok thanks sir, o actually just assumed that since they are used to represent, then they all the same. But it's ok, all take this into consideration, thanks 👍

Well sir, I don't know why you think T is different from t. T is the capital form of t. The only time we can differentiate between T and t, is when we have t and T used in two different places in the same subject. But in the Question, there is only t. So there is no reason that we should think it is different from T. Or does it goes against any rule in your place?

@@MathandEngineeringMaths and science are routinely case sensitive. And various fields have various conventions regarding what symbols should be upper and what should be lower case.

Ok thanks sir, I understand. I also just recalled that, t is the small form of T only in English language, But in math it's not applicable, thanks 👍 so much I'll adjust that

Hi friend, I am sorry I didn't get to reply to your message the other day, I was actually busy and procastinated that I'll reply later but unfortunately I forgot. To be sincere I am very happy that you like my videos, it's Quiet stressful creating them, but it gives happiness and fulfillment when someone is appreciating your work, I must say sometimes I feel discouraged but once I remember you and other friends who like th videos, I get encouraged and work toward preparing more. I have loved math right from childhood, it makes a lot of sense to me, it's just to interesting, I wonder how some people find math boring, they are great in what they love doing, everyone is perfect in doing what they are interested in. Thank you so much

And please I want to ask you, you know English is not my mother tongue, please what's the meaning of the word "procastined", is it the past tense of "procastinate", If it is then why not 'procastinated"

@@MathandEngineering my (probably correct) guess was that it was a simple misspelling of procrastinated. However, I discovered that it’s an actual obsolete variant. (From wordnik search for procrastine.) English is full of such archaic, obsolete and obscure words. My language appears to be in a race to the bottom as it were: deep, rich and subtle words and ideas are continuously replaced by the simpler ones. A few different groups will use such English words including highly educated non-natives who pick up the words while reading older texts. There are words from passages in the King James Version of the Bible that lack entries in regular dictionaries today. This is a significant problem because it becomes nearly impossible to understand older texts when words are being rapidly replaced and even redefined due to social fads, as well as philosophical, political and ideological movements. These things have effected math and science as well. Bertrand Russell addressed this problem quite forcefully in his essay “On Denoting.” He would have argued that this problem is not confined to a single language-and of course he saw its logical/mathematical implications. So sorry to give an essay instead of a simple answer. It just means “procrastinated.”

Yes thank, I like the way you answered. if you didn't explained like you did and just told me that it just means "procastinated" I would have asked for an explanation. Thank you sir

Ohh I am sorry sir, you know I have being sick for some days, I am slowly regaining my health. It was just a mistake that I wrote PS instead of ST but I have changed it now thank you for the observation. I really appreciate

@@MathandEngineering hello my friend. I hope you recover quickly and fully. I use so many of your videos in my classroom. Thank you for what you do!

I guess your approximations are not the same as mine right

@@MathandEngineering I just used the unrounded cos(80) *20 = 3.473=3.47 for BP, which leads to CP=16.53 and this discrepancy propagates to PQ by the same 0.06.

You mean?

Ok I'm sorry, it actually passed through D, this was just a mistake in the sketching, thanks for the observation

Good day sir

That's great, I'll also attempt using that

Thanks 🙏

Thank you so much sir, how are you sir?