Sweden Math | Find area of the Blue shaded Triangle | (Olympiad Math) |

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Learn how to find the area of the Blue shaded Triangle. Important Geometry and Algebra skills are also explained: similar Triangles; area of a triangle formula; Pythagorean Theorem. Step-by-step tutorial by PreMath.com
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Комментарии : 40

@poonamahalawat 3 месяца назад

First comment pls pin

@PreMath 3 месяца назад

You got it😀

@unknownidentity2846 3 месяца назад

Let's find the area: . .. ... .... ..... Here we have four right triangles: ABC, ADE, BDE and BCD. Moreover the triangles ADE and BDE are congruent. By applying the Pythagorean theorem we obtain with AE=BE and AD=BD: AE² + DE² = AD² BC² + CD² = BD² AC² + BC² = AB² AE² + DE² = AD² BC² + (AC − AD)² = BD² AC² + BC² = (AE + BE)² AE² + 2² = AD² BC² + (6 − AD)² = AD² 6² + BC² = (2*AE)² AE² + 4 = AD² BC² + 36 − 12*AD + AD² = AD² 36 + BC² = 4*AE² AE² + 4 = AD² 36 + BC² − 12*AD = 0 36 + BC² = 4*AE² AE² + 4 = AD² 4*AE² − 12*AD = 0 AE² + 4 = AD² AE² − 3*AD = 0 AD² − 4 − 3*AD = 0 (AD − 4)(AD + 1) = 0 Since AD>0, the only useful solution is: AD = 4 AE = BE = √(AD² − DE²) = √(4² − 2²) = √(16 − 4) = √12 = 2√3 AB = AE + BE = 2*AE = 4√3 BC = √(AB² − AC²) = √[(4√3)² − 6²] = √(48 − 36) = √12 = 2√3 CD = AC − AD = 6 − 4 = 2 BC = √(BD² − CD²) = √(4² − 2²) = √(16 − 4) = √12 = 2√3 ✓ Now we are able to calculate the area of the blue triangle: A(ABC) = (1/2)*AC*BC = (1/2)*6*2√3 = 6√3 Best regards from Germany

@PreMath 3 месяца назад

Excellent! Thanks for sharing ❤️

@prossvay8744 3 месяца назад

∆ABC~∆ADE Let AE=BE=a a/2=6/BC so BC=12/a in ∆ ABC AC^2+BC^2=AB^2 6^2+(12/a)2=(2a)^2 so a=2√3 BC=12/2√3=6/√3=6√3/3=2√3 So area of the Blue triangle=1/2(6)(2√3)=6√3 square units=10.39 square units.❤❤❤

@PreMath 3 месяца назад

Excellent! Thanks for sharing ❤️

@misterenter-iz7rz 3 месяца назад

Using trigonometric means, 2s cos x=6, s tan x=2, sec^2x=s^2/9=1+tan^2=1+(2/s)^2=1+4/s^2, s^4-9s^2-36=0, (s^2-12)(s^2+3)=0, s^2=12 or -3, rejected, s=2sqrt(3), thus the angle is 30°, opposite side is 6 tan 30=6/sqrt(3)=2sqrt(3), therefore the answer is 1/2×6×2sqrt(3)=6sqrt(3)😊.

@PreMath 3 месяца назад

Excellent! Thanks for the feedback ❤️

@SkinnerRobot 3 месяца назад

Brilliant, and you figured out the angle at A.

@misterenter-iz7rz 3 месяца назад

@@SkinnerRobot yes

@phungpham1725 3 месяца назад

1/ The quadrilateral DCBE is cyclic so: ADxAC = AExAB--> 6AD=a.2a= 2.sqa (1) Moreover sqAD= sqDE+sqEA= 4+sqa --> 6. sqrt (4+sqa) = 2 sqa Or 36(4+sqa) = 4xsqa.sqa Just label X= sqa, we have the equation: 4sqX-36X-144=0 Or sqX-9X-36=0 X=12 (negative result rejected)--> a = 2sqrt3 Notice that DE/AE = 2/(2sqrt3 ) = 1/sqrt3= tan 30 degrees--> The angle CAB=30 -->ABC is a special 30-60-90 triangle --> BC=AB/2= a=2sqrt3 Area of ABC= 1/2 x6x 2sqrt3=6 sqrt3 sq units

@PreMath 3 месяца назад

Excellent! Thanks for sharing ❤️

@countysecession 3 месяца назад

How do you know DCBE is cyclic?

@phungpham1725 3 месяца назад

@@countysecession because the angle at the vertexes C and E = 90 degrees, so we have a circle of which the diameter is BD.

@countysecession 3 месяца назад

@@phungpham1725 Cool. Thanks!

@LuisdeBritoCamacho 3 месяца назад

STEP-BY-STEP RESOLUTION PROPOSAL : 01) AE = BE = X 02) AB = 2X 03) BC = Y 04) (2X)^2 = 6^2 + Y^2 ; 4X^2 = 36 + Y^2 ; Y^2 = 4X^2 - 36 ; Y^2 = (2X)^2 - 6^2 ; Y^2 = (2X + 6) * (2X - 6) 05) Triangle [ABC] congruent with Triangle [ADE], so : 06) Y / 6 = 2 / X 07) Y = 12 / X 08) Y^2 = 4X^2 - 36 09) (12/X)^2 = 4X^2 - 36 10) Only Positive Solution : X = 2sqrt(3) 11) X = 2*sqrt(3) ; 2X = 4*sqrt(3) ; and Y = 2*sqrt(3). Brilliant Conclusion : X = Y 12) Checking Solutions : 13) 36 + (2*sqrt(3))^2 = 4*(2*sqrt(3))^2 ; 36 + (4 * 3) = (4 * (4 * 3)) ; 36 + 12 = 48 ; 48 = 48 14) Area of Blue Triangle (A): 15) A = (6*2*sqrt(3)) / 2 ; A = 12*sqrt(3) / 2 ; A = 6*sqrt(3) ; A ~ 10,4 16) ANSWER : The Area of Blue Triangle is equal to (6sqrt(3)) Square Units or approx. equal to 10,4 Square Units. Greetings from the University of Cordoba Caliphate (Al Andalus); The House of Greek, Arabic, Persian and Hindu Mathematical Thinking and Wisdom!

@PreMath 3 месяца назад

Excellent! Thanks for sharing ❤️ Love and prayers from the USA! 😀🌹

@quigonkenny 3 месяца назад

As ∠AED = ∠BCA = 90° and ∠A is common, ∆AED and ∆BCA are similar. Let AE = EB = x. BC/2 = 6/x BC = 12/x Triangle ∆BCA: BC² + CA² = AB² (12/x)² + 6² = (2x)² 144/x² + 36 = 4x² 4x⁴ - 36x² - 144 = 0 x⁴ - 9x² - 36 = 0 (x²+3)(x²-12) = 0 x² = -3 ❌ | x² = 12 x = √12 = 2√3 BC = 12/x = 12/2√3 = 2√3 [A] = bh/2 = 6(2√3)/2 = 6√3 ≈ 10.39 sq units

@zdrastvutye 3 месяца назад

the path to the solution can be found in the proportions of that triangle, see line 40: 10 print "premath-sweden math | find area of the blue shaded triangle" 20 l1=6:l2=2:sw=l2^2/(l1+l2)/10:r=sw:dim x(2,2),y(2,2) 30 @zoom%=@zoom%*1.4:goto 50 40 dgu1=l2/r:dgu2=sqr(abs(4*r^2-l1^2))/l1:dg=dgu1-dgu2:return 50 gosub 40 60 dg1=dg:r1=r:r=r+sw:gosub 40:r2=r:if dg1*dg>0 then 60 70 r=(r1+r2)/2:gosub 40:if dg1*dg>0 then r1=r else r2=r 80 if abs(dg)>1E-10 then 70 90 print r:l3=sqr(4*r*r-l1^2):print l3 100 la=l1:lb=l3:lc=2*r:lh=(la^2-lb^2+lc^2)/2/lc:h=sqr(la^2-lh^2) 110 masx=1200/2/r:masy=850/h:if masx run in bbc basic sdl and hit ctrl tab to copy from the results window

@srirajan1933 3 месяца назад

Professor, i love this example for 3 reasons: 1. By pivoting our view of the of the triangle to treat AC as the “base” (rather than AB), you use a known value of 6 to focus your strategy on finding BC (also its “height” from this perspective). 2. The grouping approach of reducing 576 into 2*24 helps to see how to reach 36. 3. The segment results lead to a 30-60-90 special triangle: b = 2*sqrt(3) and AC = b*sqrt(3) by 30-60-90 property. And, the longest AC side = 2a = 2*12/b from your previous derivation = 2*12/(2*sqrt(3)) = 12*sqrt(3)/3 = 4*sqrt(3) = precisely twice the length of the short side b!

@Irishfan 14 дней назад

How to do the problem makes plenty of sense, it is a beautiful thing. However the answer remains a mysterious thing until you remove the square root and show the answer in decimal form to a required accuracy for the usage of the answer!

@ИванПоташов-о8ю 3 месяца назад

When I got a notification I was listening Sabaton song.

@PreMath 3 месяца назад

Wow! Glad to hear that! Thanks for the feedback ❤️

@BKNeifert 3 месяца назад

I think the i one is the right solution. Because this isn't a real triangle. Not according to any triangle I calculated in the online calculator. I guess you have to use another method to prove your work, that's the trick and why this is an Olympiad question. I'd never be able to do that math. Maybe I'm putting it in the calculator wrong.

@mauriziograndi1750 3 месяца назад

Using hypotenuse and sides rules of triangle rectangle my result 10 unit. But all in half the time of the video. But the video answer is the one correct. Will check again.