Can you find the Height of the Triangle in 60 Seconds? | Two Simple Methods
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- เผยแพร่เมื่อ 1 ก.ค. 2024
- Learn how to find the height of the given triangle in 1 minutes using two easy methods, one being a shortcut. Step-by-step tutorial by PreMath.com
#OlympiadMathematics #OlympiadPreparation #CollegeEntranceExam - แนวปฏิบัติและการใช้ชีวิต
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So you are taking over 6 minutes to provide a solution in 60 seconds.
Another way to see it: If h meets the base at D, then triangle BDC is similar to triangle BCA.
h/a = b/c
h = ab/c
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There is another formula: 1/(h*h) = 1/(a*a) + 1/(b*b) which follows from h = ab/c. Then you can directly calculate h.
I used Pythagoras theorem to determine hypotenuse and used herons formula for area then 1/2 × base× height(here base is the hypotenuse ) then calculated the height by using both the formulas
Explanation is simple and good as well.
I'm a simple construction worker here in Switzerland. It took me around 15 seconds with the help of a calculator. With pen and paper I guess I would have managed it.
Love your channel, especially the thinking out of the box. Suddenly it seems so damn easy. It's just getting the idea that makes it so damn hard
Great Video! you've got a really good Math! I have a question. Does the formula : h= ab/c apply to all triangles with a missing height? Or does it apply only to righ triangles where we want to find the height on the other side of the triangle?
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This was a good lesson.
Really good!
I regard this as a very important, practical problem, which has all kinds of applications.
love the challenge bro, thanks for sharing both methods, great job
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I liked this puzzle. I immediately recognized the 5-12-13 triangle multiplied by 7, and realized the base was 91. I used (1/2) * 91 * h for the area of the triangle is equal to the area number produced by Heron's formula, using the lengths of all three sides. I was able to just "plug and chug" to get the value of h. I might have solved it within 60 seconds. I do not know. I probably went over time.
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How did u get the 91
@@bfoster7252 ................ The triangle is a 5-12-13 right triangle with all the sides multiplied by 7. 5 * 7 = 35. 12 * 7 = 84. The base is 13 * 7 = 91.
@@bfoster72525-12-13 ils one of the triplet of Pythagore not the most known
My method was pretty much the same as the method in the video, except that I got the formula h/a=b/c using similar triangles. As I did it in my head, I got the exact answer 420/13.
Awesome . So Clear and Easy to Follow
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you helped a lot
Thanks again prof, I am always learning something new with you. intriguing problem..
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Let D be the foot point of h. It divides side c = 91 in BD and DA with BD + DA = 91. You get 2 rectangular triangles with h^2 = 35^2 - BD^2 and h^2 = 84^2 - DA^2, or: 35^2 - BD^2 = 84^2 - DA^2. Now substitute BD by 91 - DA or DA = 91 - BD. Both give an equation with only 1 side variable, and you get for BD = 13.4615 and for DA = 77.5385. Finally h = 32.31 for one of each rectangular triangles. All just with Pyth. Theor. - I didn't know the height formula ;-)
Super, ich habe die Aufgabe durch 2 Gleichungen mit erheblich höheren Aufwand gelöst.
Angle BAC: Arctan (35/84) = 22.62 degrees. h = Sin 22.62deg x 84 = approx 32.3 units
The length of the hypotenuse can be calculated as 91 (triangle is 5-12-13 right angled triangle).
The height h = (35*84)/91 ~ 32.31 units.
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Obrigado professor. Com suas explicações fica fácil compreender. 🙏
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@@PreMath escreve muito bem em português. Parabéns.
@@luigipirandello5919 Ele usa tradutor
@@JuniorAnarchy está explicado. Obrigado.
First viewer! Wow!
You find the side length AB from pythagorean theorem and after you find the area of ABC = AB*AC/2 =h*AB
This is the solution!
Excellent problem and great solution.
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Hi,
we can find the height h in terms of a and b . Namely, h=a*SQR(1-(a/c)^2).
Have a great day!
I mean in terms of a and c.
You are soooooooooooooooooooooooooooo brilliant Man ❤❤❤❤❤❤❤❤❤❤❤❤❤
You could also solve it using similar triangles and algebraic manipulation. I did it using the hardest way, but the similar triangle method is the easiest.
Legs currently have 5 to 12 ratio, which produces a 13 hypotenuse. Drawing a perpendicular line from the hypotenuse creates two similar triangles. 35/13×12=420/13, or 32_4/13, height.
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1/h^2 = 1/a^2 + 1/b^2
this formula works for the height perpendicular to the hypotenuse and with a, b being the two shorter sides (only in a right triangle of course)
it's equivalent to h=ab/c like so:
1/h^2 = b^2 / (ab)^2 + a^2 / (ab)^2
1/h^2 = (a^2 + b^2) / (ab)^2
h^2 = (ab)^2 / (a^2 + b^2)
h = ab / sqrt(a^2 + b^2) | from the Pythagorean Theorem: sqrt(a^2 + b^2) = c
h = ab / c
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Use Euclid's Katten Theorem and you'll have the answer in less than 1 minute. both squares over the squares give the square over the hypotenuse. since we know the area of the square above the minor square, all you have to do is divide the area by the length of the hypotenuse and you get the length of the minor square of the triangle that WILL be formed with height. so you have two lengths. the rest is done, the pythagorean theorem.
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Most welcome
We can use similarity method also to find height
The first part is to find the area of the triangle using Heron’s Formula*.
For a triangle with side lengths of a, b, and c.
Area = square root of (s * (s-a) * (s-b) *(s-c))
The second part is: The height = (2 * Area)/whichever side you want as the base
*Proofs for Herons Formula are easy to find on the web.
After I recognized the 5, 12, 13 triple and got the length of the hypotension as 91. I just made the triangle into a rectangle to get the area of the rectangle and divided by 2 to get the area of the triangle. The area of the triangle was 1470 units
So now, A = 1/2 b x h.
1470 = 91 h / 2
2940 = 91h
H = approx 32.31
I have a question , besides from the formula is their any other simpler formula ?
I would prove ab = hc geometrically:
ab = rectangle with sides a and b (composed of four triangles, twice the one on the left and twice the one on the right of the height)
hc = rectangle with sides h and c (composed of four triangles, twice the one on the left and twice the one on the right of the height)
You have simplified the sides of the triangle by a factor of 7 to the Pythagorean triple 5, 12 and 13. This gives the height h' = 5 * 12 / 13 = 4.62. Multiply 4.62 by 7 and you get h = 32.31.
Is the second method using a right angled triangle forms 1/2 a rectangle
Absolutely beautiful...the proof is Asymmetric !🙂
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Another method, in right angle triangles when a line is draw perpendicular from the base up to angle C, the sides are all proportional
Side AB has a ratio of 2.6 to side 91. Using this you can conclude the short side of the smaller triangle is 13.461, using pythagorus theorem you can get the remaining side.
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Could we find c and then using trigonometry find h
Using Pythagoras Theorem we get AB as 91 units. Area of triangle ABC = 0.5(35)(84) = 1470 units^2. Then area of triangle ABC = 0.5(AB)(h) = 1470. 1470(2)/91= 32.31 units^2.
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Très intéressant ...like usual . thx
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First time that i see this methods.
Just find the triangle area and the hypotensus then 1/2×bh 1470 so b×h÷2=2940/91=32. 307
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I simply say you r such awsom
Master sir.. Thank you sir
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Hi, I just used to formula Height= 2*A/B where A is the area and B is the base
Note: when using this formula the height is the one perpendicular to the base so in this formula 91 will be the base
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One more easy way is that find its area by heron's formula and and compare the value with the formula of triangle which is 1/2×b×h and then find h easily 👍
its a 5:12:13 triangle times 7 so AB=7•13=91. A=bh/2 we have 2 different bases and heights so 35•84/2=91h/2. 91h=2940. h=420/13
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There is another method. You can apply the pythagorean theorem to ABD and ACD where AD=h and you find the high as a function of a square root.
And there is another method. You can use similar triangles ABD and ACD and you find the high as a function of square root again.
Thanks teacher! 😀😀😀💯💯💯❤❤❤❤❤❤
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My methods too.
Because the original triangle with perpendicular sides 84 and 35 is a 12/5/13 triangle enlarged 7x, all you need is to do ratios. The triangles are similar therefore the hypotenuse 35 is to 13 as the height h is to 12. Therefore h is 35x12/13. All the rest of this video is padding.
Before watching: BA = 91 by Pythagoras. The area of the right triangle is 35*84/2=91*h/2. This leads to h being 32.31
After watching: approaches 1 and 2 are actually the same, approach 2 is just the derivation of the formula used in approach 1.
Using some Trig. will also solve: Find angle A by the using Sine, SineA = 35/91= Sine(0.384) = 22.6degree... therefore angle B = 90 degree - 22.6 degree = 67.4 degree....
h/35 = Sine(67.4 degree), h = 35(Sine 67.4 degree) = 35 x 0.923 = 32.31
What if it's not a right triangle? Is it stil possible to find the H ?
You can do it without having to calculate c. 1/h^2 = (1/a^2) + (1/b^2)
Thanks for video. Good luck!!!!
Excellent question sir!!!!!
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Why when using herons formula different answers?
I solved it in two ways.
The second one that you show on the video and using the three rectangle triangles.
(i) h²+x²=35²
(ii) h²+y²=84²
(iii) 35²+84²=(x+y)²
Now, (i) - (ii) --> 84²-35²=y²-x² --> 5831=y²-x²
Doing the square root of (iii) --> |x+y|=91 ; We only need positive solution, so: x+y=91 --> y=91-x
By sustitution in (i) - (ii):
5831=(91-x)²-x² --> 5831=8281-182x+x²-x² --> 182x=2450 so x=2450/182, simplifying --> x=1225/91
Finally, by sustitution in (ii):
h=√[35²-(1225/91)²]≈32.3 units
P.S. Excuse my English
32.3077 approximately. So, easy question .
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This is actually 5,12,13 triplet if you divide by 7
Then Area is 0.5 x 5 x 12 = 30
So Area is also h x 0.5 x 13 so h for this traingle is 2 x 30/13. Multiply by 7 again it's 420/13. Did it in my mind! 😌
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superb!
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With a calculator I would simply find the angle BAC using arching 35/84 then calculate h with h=sin BAC × 84
I still haven't watched the vid
So I first of all used pythagoras to find the hypotenuse of the large triangle
Which gave me 91
Then use the cosine rules to find the angle
Then used trig ( sin function ) to find the side giving me 32.3
Another nice approach! Trig.
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great video
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Thank
I got the solution using trigonometry:
Berry berry good
My school expects us to learn this even though we haven’t learned pythagorias or square root
Is this way useful with all triangle or only right angle?
only right angle
Oh i see my brute force strategy was correct, but i was so confused when i saw the height i thought theres no way they gonna make me do such a hard question in grade 7
I dont get why the two areas on the second methid shoulg be equal. From the image (a*b)/2 is clearly larger than (c*h)/2. Actually the first triangle contains the second one within, where does the equallity come from
h=420/13≈32,31
Find the area of triangle frist.Secondly find the hypothesis of the triangle AB. This can find height h .
I was told there is an old Indian method predated to Pythagorean. If you want to find the hypotenuse, you can add 7/8 of hyp's horizontal side and 1/2 of hyp's perpendicular side. In this case BA = 7/8(84)+1/2(35); BA = 73.5 + 17.5; BA=91
I was blown away when I learned a^ + b^ = c^ 25 years ago.
Now I'm equally amazed, if not more so by this method you have presented here.
I sure do thank you for taking the time offer this awesome equation.
This only works on certain right triangles, such as the 5:12:13 type (scaled by k) in this video:
(k*5)/2 + (k*12)*7/8
= k*5/2 + k*21/2
= k*26/2 = (k*13)
It also works for the 3:4:5 triangle:
(k*3)/2 + (k*4)*7/8
= k*3/2 + k*7/2
= k*10/2 = (k*5)
It fails on other types. In fact, only the first 2 entries of the first 10 in the table work:
3:4:5 3/2+7/2=5
5:12:13 5/2+21/2=13
7:24:25 7/2+21=49/2 ≠ 50/2
8:15:17 4+105/8=137/8 ≠ 136/8
9:40:41 9/2+35=79/2 ≠ 82/2
11:60:61 11/2+105/2=58 ≠ 61
12:35:37 6+245/8=293/8 ≠ 296/8
13:84:85 13/2+147/2=80 ≠ 85
16:63:65 8+441/8=505/8 ≠ 520/8
20:21:29 10+147/8=227/8 ≠ 232/8
A third, not quite as simple, solution:
tan(BAC) = 35/84 ; BAC = 22.62deg
The big triangle and the small one to the left of h are both right triangles with ABC in common, so the angle between a and h must also be 22.62deg.
cos(22.62) = h/35 ; h = 32.31
h=32(4/13)unit.ans
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Dear Ayoub, we use Camtasia techsmith utility along with the photoshop. Thanks for asking😀
My first impression was that Trigonometry is easier: TanA = 35/84= 22.62 degrees, Sin22.62 = h/84 = 32.31
This is a similar triangles problem where the scale factor is 35/91 on side lengths. Use that to scale the side of length 84, and you immediately get (35/91)*84. giving the correct answer.
Thanks for sharing this information. I see that it goes both ways ie.
84/91 x 35 = same.
I know a^ + b^ = c^, but I saw a comment I'll share w/U. Only this one doesn't work both ways. I didn't try to correct him/her about describing side as perpendicular to hyp, I just thanked them for posting.
I was told there is an old Indian method predated to Pythagorean. If you want to find the hypotenuse, you can add 7/8 of hyp's horizontal side and 1/2 of hyp's perpendicular side. In this case BA = 7/8(84)+1/2(35); BA = 73.5 + 17.5; BA=91
@@simpleman283 i tried that 7/8 method in another problem it gave wrong answer
@@70newlife Yea, me too. But you've got to admit if it predates Pythagoras, it's pretty cool.
good problem!!!! i found it to be 420/13 which is the same as your decimal answer
I used another method. First with Pythagorean theorem I knew AB = 91. Second I excel what CD = h, and excel what BD = x. So AD = 91 - x. So I wrote using Pythagorean theorem what
35^2 - x^2 = 84^2 - (91 - x)^2
1225 - x^2 = 7056 - 8281 + 182x - x^2
182x = 2450 | : 182
x = 175/13
Then I contained numbers and I got what h ≈ 32,31. So is the same answer, but another method.
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Another explanation of h = ab/c is that. Sin Inv Angle CAB = a/c Sin Inv Angle CAB = h/b So h = ab/c
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@@PreMath 💟💟💟💟
How h=(ab/c)? What if i rotate the triangle? a,b,c will remain same but how h will change?
Dear Mohsin, just keep in mind as long as the side "c" is the hypotenuse and "h" is the height of the big triangle. The result would be same. Hope I explained well.
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Since ab/2 = ch/2 reduces to h = ab/c, aren’t these the same method with extra steps?
The "2nd method" was not so much a new method, rather it was to justify the formula used in the 1st method.
A= 35x 84 ÷ 2 =1,470
I solved it by another way
By getting the area of the triangle
looking at the smaller triangle i see a 3,4,5, triangle. 35/5=7, and 7x4 =28???????????
12x5/13 x7=420/13.
V nice .
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@@PreMath l have learnt so much from your channel.l watch your old videos regularly
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@@PreMath 🙏🙏Your way of teaching is amazing.You go to the minutest detail of every question which others lack.Even a novice understands your soln
Dang got it wrong. I just stood a up straight and compared it to H and called it 32
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The two methods are the same. The path of the second is the reverse of the first and that's it.
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It's nice
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A=1,470