30-60-90 triangle side ratios proof | Right triangles and trigonometry | Geometry | Khan Academy

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Proving the ratios between the sides of a 30-60-90 triangle
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[@sdhafele5003]

If you draw an isoscolese triangle with the 2 congruent sides labled as "a," the triangle's height, h, will split the base, b, into 2 parts. If one of those parts is x, the other MUST be b-x. By the Pythagorean Theorem, x^2+h^2=a^2. By the Pythagorean Theorem, (b-x)^2+h^2=a^2. Using the Algebraic Identity (a-b)^2, you'll now know that b^2-2bx+x^2+h^2=a^2. Replacing x^2+h^2 w/ a^2 gives you b^2-2bx+a^2=a^2. Solving this equation for x tells you that x, one of the base's 2 parts, is b/2. The other part of the base, b-x, is b-(b/2), or, (2b/2)-(b/2), or simply b/2. All this proves that the height of an isoscolese triangle cuts the base of the triangle in half.

[@lukababilodze4283]

Thank you my math teacher doesn’t know how to teach us properly. (Of course she has the knowledge tho)

[@user-ux2gz7sm6z]

I doubt she has the knowledge xD

[@lukababilodze4283]

@@user-ux2gz7sm6z lol I just saw a reply on my comment. I don't even remember commenting this. At least now I know that she does not have the knowledge XD I was in probably like 9th grade at the time

[@zurabkhutsishvili8754]

@@lukababilodze4283 :DDDDDDDDD mecxre klasshi tu ameebs uyurebdi sagol shen dzmao

[@MikeCasey311]

Thank you, thank you. Every other video used these values without proof. 👍👍🇺🇸

[@MoonLight11023]

This theorem can be proven by inscribing a 60-60-60 triangle (ABC) in a circle. Draw AB line such that it makes an angle of 30 deg with the horizontal diameter of a circle, drawn in a Cartesian coordinate system, of a center point (0,0). Then, draw BC line such that it makes right angle with the horizontal diameter and then draw AC line. Since BC line is perpendicular to the horizontal diameter, therefore, BC line is bisected by the horizontal diameter in, let's say, D point. Continued.....

[@MoonLight11023]

If we look closely we will find out that two congruent right triangles are formed ADB and ADC. As a conclusion, BD line = DC line = 0.5*BD line. ABD angle = ACD angle = 60 deg. DAB angle = DAC angle = 30 deg. It's concluded that ABC is a 60-60-60 triangle. The inscribed angles, BAC, ABC, and BCA are equal to 60 deg and they are equal to half of their central angles each. Sector length = radius* angle(rad). Since the three angles of the inscribed 60-60-60 triangle are equal, therefore,Continued..

[@MoonLight11023]

therefore, the length of the sectors AB, BC, and AC are equal. It's concluded that AB line = BC line = AC line. BD line = CD line = 0.5*AB line = 0.5*AC line. I hope it helps :)

[@standardservices7573]

this really helped in my exams thanks

[@ThomasGiles]

I think I get it now. Took a a lot of re-watching. Towards the end, you were moving a bit fast, I think.

[@trentonbouchard988]

Perfect. I needed this video. I finally get it

[@jonathan8114]

Where on earth did he get all those 4s, he just kept on adding them out of nowhere, someone please explain?

[@jordieb3333]

fr lol but he got x^2/4 from simplifying (x/2)^2 but the 4x^2/4 i have no idea

[@Jamey-xm1dd]

He just made X^2 into 4x^2/4 to get a common denominator, making it easier to subtract x^2/4 from 4x^2/4 which equals 3x^2/4.

[@natcat1324]

It is easier to think about to have the short leg as x, the long leg as x√3, and the hypotenuse as 2x.

[@jordieb3333]

Please explain madam, i have a test in IM3 and i dont understand this man he did the pythagorean theorem incorrectly and wrote a random 4 at the end and my mind is having difficulties

[@lenoraelaine]

That was fun!

[@Ensign_Cthulhu]

@Veronicaboost You can never watch too much Khanacademy before a test. Provided of course that it's relevant to the test...

[@katrinaultriyama7143]

can you help me with this problem? "The Pythagorean theorem is used to prove the side lengths of a right triangle. Use complete sentences to describe the proof of a right triangle with sides lengths of 7 meters, 24 meters, and 25 meters."

[@palereaper]

no idea man

[@jordieb3333]

im 3 yrs too late, but its saying use the pythagorean theorem to prove why that right triangle has side lengths of 7m, 24, and 25m and detail in a explanation . so just give an example, for example lets say 25 was x, you would write down your use of the pythagorean theorem(A^2+B^2=C^2) to get 25 its actualy pretty simple if you dissect the question a bit. I know it wont help but maybe this will help another kid looking for answers.

[@skateraptor12]

How is the leg bigger than the hypotenuse then when the hypotenuse is 10 radical 3? That means that 10 radical 3 times radical 3 all divided by 2 is 15?

[@ThomasGiles]

Why are we multiplying and voiding by 4

[@dildobaggins2759]

"Khan academy ime gonna drop an altitude". Me: "Just say you drew a line"...

[@elmosarmata9561]

It's the correct mathematical term tho.

[@dildobaggins2759]

@@elmosarmata9561 Yeh but more people will understand draw a line you want to make maths as simple as possible to people when you teach. Or anything in general no need to use complex terms for no reason.

[@thecosmos7671]

Altitude is more appropriate because a line could project any angle at the opposite side but an Altitude projects a 90° angle and that makes it easy for us to use Pythagorean theorem and also in an equilateral triangle a median is an altitude. A median is a line drawn from the vertex to the opposite side which bisects (divides in 2 equal parts) the opposite side Hope this helps

[@bofa-zi4fj]

why is (BD)^2 not simplified to B^2 D^2?

[@suadali1017]

Thank you so much .

[@novanova4737]

just draw an right angled triangle then construct another right angled triangle then prove the whole triangle to be equiangular triangle then start the proof it is the easiest method

[@afkoppauthor]

Some people can't get it as fast as others though. Everyone has a different way of thinking.

[@aneeshsunny7274]

Well nice cud xplain in a better way

[@LordOfNoobstown]

why can't you take the square root first in your final proof on x sqrt(3)/2

[@ninderchitosiya5326]

Sr hindi me post krna plz ya hindi medium ka koy channel btana plz

[@GabrielOrtiz-dv4jk]

What about the square 3??

[@babyhppoo5763]

What's that weird thing they have thats Like the square root of three and stuff

[@shridhar1286]

your genius

[@faridhussainmirza4524]

nice video

[@lifer7741]

Someone explain how that 4x²/4 turned into 3x²/4

[@thecosmos7671]

It didn't turn into √3 s^2/4 Acc to Pythagoras Theorem we have a^+b^2=c^2 Here,we know C and b so to get a we write C^2-b^2=a^2 C=X, b=X/2 So , x^2-(X/2)^2=3x^2/4 On taking square root we get √3x/2. I hope it helped

[@lifer7741]

@@thecosmos7671 yes i understand, thank you. But wait how did x²-(x/2)² equal to 3x²/4?

[@thecosmos7671]

@@lifer7741 Okay let me explain. You can write x^2-(x/2)^2 as x^2-(x^2/4) okay. Now, we'll make their denominators equal so that we can do subtraction So we'll make x^2 =4x^2/4 (multiplying and dividing by the same number preserves the equality,means it does not change the original value ,its just easier for us to compute, otherwise we'd have to take LCM. well it's the same as taking LCM but I guess it comes more easily) So , 4x^2/4-x^2/4 =3x^2/4 ( if it's not clear here then suppose x^2/4 is a mango So 4 mangoes-1 mango =3 mangoes ( now put the value we assumed to be a mango in the result.) Which means 3 *x^2/4 I hope I helped ⭐

[@brooklynnmiller8446]

HOW THE FUCK DID YOU GET A FOUR!!!!

[@josiahm5221]

He simplified (x/2)^2. 2^2=4 and x^2=x^2 making x^2/4

[@julescallanta6539]

My teachee told m that the opposite of the 30° was always x and the opposite of 60° is x/3 and the other is 2x

[@shahdrashed5938]

Math proplems in exam paper: Solve me!? Me. Lol.

[@tuckersawyer4414]

why do you have x/2 in this video for the short sides but in reality its x. why?

[@afkoppauthor]

It depends on who teaches it. It all depends on if the hypotenuse is 2x (when the short side is x), or if the hypotenuse is x (when the short side is x/2). it all depends on the teacher. :)

[@NarcoticNKO45]

@dere60 ikr

[@cinnamonbeanz2620]

This still makes no sence to me.

[@kabascoolr]

Watch his earlier video on isosceles triangles, and the one on Pythagorean theorem. Most of the ideas for this video comes from those.

[@novanova4737]

meaningless video just made difficult for students

[@elmosarmata9561]

Just learn lmao

[@lastravaganza2385]

Just learn lmao