Can You Find the Area of the Blue Square in the Right Triangle? | Fast & Easy Tutorial
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- เผยแพร่เมื่อ 10 ก.พ. 2025
- Learn how to calculate the area of the shaded square DEFG which is inside of a right triangle. Step-by-step explanation by PreMath.com
تمرين جميل جيد . شرح واضح مرتب . رسم واضح مرتب . شكرا جزيلا لكم والله يحفظكم ويرعاكم ويحميكم جميعا . تحياتنا لكم من غزة فلسطين .
I think it's easier than all that. Indeed : all the right triangles formed are similar and the problem can be solved by simple proportionality between their sides.
Let us call "S" the side of the square and divide the small leg of the larger triangle ( which is worth 39 ) into 2 segments : the upper one will be "a", and the lower one will be "b", being clear that a + b = 39.
We can thus form a simple system of 2 equations with 2 unknowns :
65/39 = S/b
65/52 = a/S = ( 39 - b )/S
And solving it, we find that S = 780/37...so the area ( S squared ) will be approximately 444.41 u2
Thanks. ..and greetings from Spain. .
A simpler alternative to approach this example is to notice that x/(52/39) + x + x/(39/52) = 65. Solving the linear equation we obtain x=780/37 and the rest is history!
At 1:25
The complementary angles are all the same in the four triangles hence all FOUR triangles are similar with ratios 39,52,65 that is 3,4,5
Let x=length of side of square EDGF,
Consider triangle AEF,
AE,EF,FA >>>>> 3,4,5 >>>>> 3x/4,4x/4,5x/4
AF=5x/4
Consider triangle FBG,
FB,BG,GF>>>>> 3,4,5 >>>>>>3x/5,4x/5,5x/5
FB= 3x/5
AB=AF+FB=39
39=5x/4+3x/5
multiply by 20,
780=25x+12x
37x=780
x=780/37
Area of blue square EDGF=x^2
=(780/37)^2= 608400/1369=444.41 square units approx.
This is a good place to stop.
Note that Professor actually gives you more information than you need, which is unusual. Because you can get the 65 from the Pythagorean theorem. It's nice when problems like this are set up with things like 3-4-5 right triangles because then the effort goes into problem-solving rather than having to get bogged down in arithmetic.
Great example again thank you 👍🏻
Amazing video😊👍 thank you so much sir
شكرا لكم من المغربBH=h
هو ارتفاع ABCو×-h هو ارتفاعBGF
نجمع مساحات الأجزاء الداخلية لABC نجد مساحة ABC بعد حل معادلة من الدرجة الأولى نجد X=780/37
Very good explanation of solution of the given geometrical problem
Thank you so much Ali for your continued love and support. Take care dear and stay blessed😃 You are awesome. Keep smiling😊
Beautifully explained keep it up
Faster way starting from BFG (parallel ipoth). You can see that AF + FB = 39 so 5/4x + 3/5x = 39 so x=39*60/37 and A=x²
Thanks Roberto for the feedback. You are awesome 👍 Take care dear and stay blessed😃
@@PreMath good work tho!!!
very nice - the power of similar triangles!
very well done, great job bro
I got that one. It was a bit challenging. Give more like this..
Nice explanation but while calculating value of x plz check 780=9x+12+16x u forgot to write x with 12 which is confusing but rest is OK thanks for this calculating method
Thanks Raj for the visit! You are awesome 👍 Take care dear and stay blessed😃 Kind regard
@@PreMath ji thanks for reply but I wanted to draw ur attention towards mistake in equation by chance ,plz take care in future like it 780=9x+12x+16x but u had written 780=9x+12+16x (forgot to write x with 12 but speaking)
@@RajKumar-ci7dg I was confused also. Good comment.
@@jeffweber8244 ji it was by mistake, he/she forgot to mention x ,as replied any how it's human error
Thank you for premath
The right triangle ABC is similar to the well-known triangle 3:4:5 and the similarity ratio is 13 → If we analyze the triangle 3:4:5 and “X”= Side of the inscribed square → AB=3=AF+FB → The interior triangles are similar to ∆ABC ⇒ AF=5X/4 and FB=3X/5 ⇒ 3=(5X/4)+(3X/5) → 37X/20=3 → X=60/37 ⇒ X²=3600 /1369 → Area of the square inscribed in the proposed triangle = 13² X² = 13²(3600/1369) = 444.4119
So very interesting, keep up the good work, love from Germany.
I was able to do this myself. Yayyy!
thank you very much, from Mrrocco
Thank you teacher
If length of each side of this square be 'a' then the sides of the smaller triangles are (a, 3a/5, 4a/5).
(5a/3, a , 4a/3)
and (5a/4, 3a/4 , a)
Hereby 4a/5 + 5a/3 = 52
or a = 52*15/37
3a/5 + 5a/4 = 39
or a = 39*20/37
I'm assuming BG as a and comparing triangle BGF with triangle ABC, and triangle CDG with triangle ABC, got the same answer.
There should be another ways to get the answers by comparing similar triangles twice.
Thank you sir
Very nice vídeo. Professor, have a happy week, with peace and Health.
Likewise dear
Thanks Luis for the visit! You are awesome 👍 Take care dear and stay blessed😃 Kind regards
Is good.
think you .
There was already a similar task, the Egyptian triangke was slightly disguised
Mathematics is the science that deals with the logic of shape, quantity and arrangement. Math is all around us, in everything we do. ... The needs of math arose based on the wants of society. The more complex a society, the more complex the mathematical needs.
It was easy to solve, but the value of the answer was very half-hearted, so I was very worried if I was right or not.
i like the video
The triangle BFG is also similar to the others
Thanks for the visit! You are awesome 👍 Take care dear and stay blessed😃 Kind regard
It's always the algebraic parts that are most tedious, it seems.
13*(3,4,5)=(39, 52, 65). So this triangle should be right.🙂
yes
Correction: 780=9x+12+16x is wrong, right is: 780=9x+12x+16x !!!
#Pythagoras #PythagoreanTheorem #PythagorasTheorem
Why Pythagoras? ...Read my answer. ..and learn.
Alpha + Beta is 90 degrees.
🤔
Pythagorean Theorem
Not at all. Read my answer.
I can't solve this without your help🤔✨✨
Are you sure ?....
@@alvarosarmientogomez4996 Yes 😌✨✨
Wrong 39x 25x+12
Pythagoras
Not at all. Read my answer. ..and learn.
I don't understand
First
Wrong 39x 25x+12