Can you find area of the Purple Triangle? | (Important Geometry skills explained) |

[@mohanramachandran4550]

Very interesting & Simple logical sum

[@PreMath]

Glad you think so! Thanks for your feedback! Cheers! 😀 You are awesome. Keep it up 👍

[@aleksandrsavuskan7124]

S=567

[@michaelkouzmin281]

Triangles AEF and ABC are similiar, so we can either calculate coefficient of proportion (22/8) and multiply 56 times square of this coef or calculate AB and then Area of ABC = AB*CB/2

[@jphilsol6459]

yes I did the same, I think is faster.

[@IslamDarbal]

Me too

[@marioalb9726]

Blue square, side 'b' b = √Area = √196 b = 14 cm Yellow right triangle, height "h' Area = ½ b. h = a h = 2 area / b = 2 . 56 / 14 h = 8 cm Height of purple right triangle: H = b + h = 14 + 8 H = 22 cm Areas ratio of similar right triangles R = A/a = (H/h)² R = (22 / 8)² = 2,75² Area purple triangle: A = R. a A = 2,75² . 56 A = 423,5 cm² ( Solved √ )

[@ybodoN]

Generalized: the area of the purple triangle is _(a + 2b)² / (4b)_ where _a_ is the area of the blue square and _b_ is the area of the yellow triangle. The area of the white triangle is _a² / (4b)_ so the area of the purple triangle is _a + b + a² / (4b)_ i.e. the famous identity _(a² + 4ab + 4b²) / (4b)._

[@tombufford136]

A quick glance at the problem. Height h = sqrt(196)= sqrt(56) .length, L = h/cos(45). h = 21.5 and L =30.4. Purple area = half base x height = 15.2 * 21.5 =326.8. I look forward to watching the video.

[@marinacheckina2588]

Откуда взялся угол в 45 градусов?.

[@tombufford136]

Hi Marina , I corrected my original comment soon afterwards. the angle required CAB is 33.7 degrees To solve this without angles , similar triangles are used, Triangles EAF and CAB are similar triangles with the same side ratios. Here is my revised comment from a month ago :-" Correcting more accurately. height =sqrt(196)+ 56/(sqrt(196)*0.5)=14+8=22.Angle at A = ATan(8/14) =33.7 degrees. L=h*14/8=38.5. half base * height =19.25*22=423.5"@@marinacheckina2588

[@tombufford136]

Привет, Марина, вскоре после этого я исправил свой первоначальный комментарий. требуемый угол CAB составляет 33,7 градуса Чтобы решить эту проблему без углов, используются аналогичные треугольники, Треугольники ДСП и CAB представляют собой аналогичные треугольники с одинаковыми соотношениями сторон. Вот мой исправленный комментарий месячной давности :-" Поправляю точнее. высота =sqrt(196)+ 56/(sqrt(196)*0.5)=14+8=22.Угол в точке A = ATan(8/14) =33.7 градусов. Л=ч*14/8=38,5. половина основания * высота = 19,25 * 22 = 423,5 дюйма @@marinacheckina2588

[@advancedintention7169]

Thanks sir....

[@KAvi_YA666]

Thanks for video.Good luck sir!!!!!!!!!

[@AmirgabYT2185]

S=423,5 cm²

[@bigm383]

Nice work!❤🥂

[@PreMath]

Thank you! 😊

[@thewolfdoctor761]

I calculated area of purple triangle = sum of areas of blue square, yellow triangle, white triangle so, 1/2(22*X) = 196+56+1/2(14*(X-14)) ==> X= 38.5 ==> Purple area = 1/2(38.5*22) = 423.5

[@yalchingedikgedik8007]

Thanks Sir We enjoyed with like these methods of solve .

[@zipponvr7043]

You can solve it much simpler, you don't need to calculate any sides 196 / 56 / 2 = 1.75 it is acpect ration EA and EF ( EF = 1.75 x EA ) EFA and FGC are similar, so area of GFC = 196 * 1.75 / 2 = 171.5 that's mean area DAC = ABC = 171.5 + 196 + 56 = 423.5

[@HeHe-rp3dq]

It can be done with trigo. We just need to find tanDAC and then equate for the bigger triangle DAV and then simply apply the formula to get the ans.

[@williamwingo4740]

Here's a different, and possibly simpler method: DE = √196 = 14 (done by inspection: it has to be between 10 and 15; I tried 14 since 14^2 has to end in a 6--and it worked); EA = (2)(56)/14 = 56/7 = 8; Triangles EAF and GFC are similar by angle-angle-angle; so GC = EF(14)/8 = (14)(14)/8 = 196/8 = 24.5; Area of triangle GFC = (0.5)(24.5)(14) = (24.5)(7) = 171.5; line AC bisects the big rectangle; so the area of the lower purple triangle is equal to the sum of the upper three triangles: 196 + 56 + 171.5 = 423.5. Cheers. 🤠

[@sie_khoentjoeng4886]

Make a line FG cross to AB at H, then area ractangular AHGD is 196+56+56=308. Since DG=√196=14, then AD=308/14=22=BC , and AE=FH=22-14=8 AHF ~ABC, then AB/BC=AH/HF X/22=14/8 or AB=22*14/8=38.5 Finally area of ABC = 0.5x22x38.5=423.5 unit..

[@santiagoarosam430]

EF=GF=√196=14 → EA=2*56/14=8 → s=14/8=7/4 =Razón de semejanza entre ∆FGC y ∆AEF → Área ∆FGC=56s²=171.50 cm² → ∆ADC=∆ABC → Área púrpura =196+56+171.50 =423.50 cm² Gracias y saludos.

[@SaadmaanMahmidShoshi]

Solved it in a bit of a lengthy way - calculated one side length of the rectangle which was 22, then calculated area of the white triangle at the right, which came as 7a-98 (assuming the other side of the rectangle as a). Then, the area of the bigger triangle became 0.5× a × b = 0.5 × a × 22 = 11a. Now, summing the areas of the blue square, the yellow rectangle and the white triangle, we get the total area = 154 + 7a, which is equal the area of the big triangle. From there we get a = 38.5. Now we can simply plug in the values of a and thus we get area of bigger triangle = 11a = 11 × 38.5 = 423.5. That is indeed the area of the purple triangle, as the diagonal of a rectangle bisects it into two identical triangles of same area. 😊

[@jimlocke9320]

Triangle ΔAEF and ΔABC are similar by angle-angle (

[@wackojacko3962]

@ 0:55 , Gotta love those Action plans: because we as mathematicians need a definitive checklist of tasks and resources too achieve a goal...and in this case the area of the Purple Triangle. If you watch the video till the end you'll find 2 Action Plans ....and we know math is tricky and that there's mo ways to skin a cat! AI uses optimization algorithms to minimize and maximize too get better results. In this case the results are the same. I think the result using the Area of a Quadrilateral is better. 🙂

[@PreMath]

Great! Thanks for your feedback! Cheers! 😀 You are awesome. Keep it up 👍

[@quigonkenny]

Let s be a side of the blue square, h be the height of the yellow triangle, and b be the base of the purple triangle. Blue square DEFG: A = s² 196 = s² s = √196 = 14 Yellow triangle ∆AEF: A = bh/2 56 = 14h/2 = 7h h = 56/7 = 8 As ABCD is a rectangle, with right angles at every corner, by Complementary Angles ∆AEF and ∆CBA are similar. AB/FE = CB/AE b/s = (s+h)/h b/14 = (14+8)/8) = 22/8 = 11/4 b = 14(11/4) = 77/2 Purple triangle ∆CBA: A = bh/2 = (77/2)(14+8)/2 = (77/2)22/2 A = 11(77/2) = 847/2 = 423.5 cm²

[@murdock5537]

∎ABCD → AB = AH + HB = 14 + (x - 14) = CD = DG + CG = 14 + (x - 14) AD = AE + DE = (y - 14) + 14 = BC = BK + CK = (y - 14) +14 area ∎DEFG = area ∎BKFH = 196 → DE = EF = 14; area ∆ AFE = area ∆ AHF = 56 14y = 308 → y = 22 → y - 14 = 8 → 8(x - 14) = 196 → x = 77/2 → area ∆ ABC = xy/2 = (7/2)11^2

[@nhlalonhleduma8034]

Do we use that formular on any triangle

[@tombufford136]

Correcting my previous answer. Height h=sqrt(196)+sqrt(2*56). Length L = h/cos(45). h=24.6, L=34.8. half base * height = 17.4 * 24.6 = 427.6.

[@MrPaulc222]

I went for the portion of the purple triangle on the left and wrote 8/14 = 22/x which gave x as 38.5. Therefore, area of purple triangle is (22*38.5)/2 so 11*38.5 sq cm.

[@Waldlaeufer70]

56 / (196/2 + 56) = 56 / 154 = 28 / 77 = 4/11 A = (11/4)² * 56 = 121/16 * 56 = 121 * 3.5 = 423.5 cm²

[@raya.pawley3563]

Thank you

[@Copernicusfreud]

Yay! I solved the problem. Although the yellow and purple triangles are inverted in the diagram, the yellow triangle is also similar to the purple triangle.

[@じーちゃんねる-v4n]

EF=14 EA=8 ∠EFA=∠CAB=θ then tanθ=8/14=4/7 BC=22 AB=22/tanθ=77/2 ∴S=11(77/2)=847/2

[@kennethstevenson976]

Got the answer on the first try, step by step with diagram.

[@e1woqf]

The triangles AEF and ABC are similar. We just need to find the scaling factor, which is 22/8.

[@GorkiCiprian74]

Also purple trianle is congruent to yellow one.

[@AndreBaguette]

I used ratios

[@alster724]

Easy even without the trapezoid

[@tombufford136]

Looking more carefully, I have made the mistake of calculating the 56 unit area as a rectangle not a Triangle , making my answer inaccurate.

[@giuseppemalaguti435]

14/8=b/22....b=22*14/8=77/2...A=77/2*22/2=11*77/2=847/2

[@marioalb9726]

Blue square, side 'b' b = √Area = √196 b = 14 cm Yellow right triangle, height "h' Area = ½ b. h h = 2 A / b = 2 . 56 / 14 h = 8 cm Height of purple right triangle: H = b + h = 14 + 8 H = 22 cm Similarity or triangles: B/ H = b / h B = b. H/h B = 14 . 22 /8 B = 38,5 cm Area purple triangle: A = ½ B.H A = ½ 38,5 .22 A = 423,5 cm² ( Solved √ )

[@danieldennis9831]

⇒A of △ABC=423.5 cm² □DEFG has A 196cm² which is 14². Each side is 14 cm. |DE=14 |EF=14 △AFE has A 56cm². A of △=½bh. |AE=x 56=½(14)x. 112=14x. x=8 |AE=8. |AD=|AE+|ED=8+14=22. I could but will not include all steps to prove △AEF△FGC△ACD are all similar and therefor have congruent angles. tan∠EAF=tan∠DAC. |DC=y 14/8=y/22 y=38.5 With △ADC having base 38.5 and height 22, A=423.5 △ADC is mirror of △ABC and is therefore △ADC≅△ABC. A of △ABC=423.5cm²

[@misterenter-iz7rz]

not difficult, by first impression. AE=2x56/14=8, GC=14x(14/8)=49/2, therefore the answer is (1/2)(14+49/2)(14+8)=11(28+49)/2=11x77/2=423.5.😊

[@PreMath]

Excellent! Thanks for sharing! Cheers! You are awesome. Keep it up 👍

[@gelbkehlchen]

Solution: DE = EF = side of the blue square = √(196[cm²]) = 14[cm] AE*14[cm]/2 = 56[cm²] |*2/14[cm] ⟹ AE = 56[cm²]*2/14[cm] = 8[cm] ⟹ BC = AD = AE+DE = 8[cm]+14[cm] = 22[cm] ⟹ Similarity: AB/BC = EF/AE ⟹ AB/22[cm] = 14[cm]/8[cm] |*22[cm] ⟹ AB = 14[cm]/8[cm]*22[cm] = 38,5[cm] ⟹ Area of the Purple Triangle = AB*BC/2 = 38,5[cm]*22[cm]/2 = 423,5[cm²]

[@aka_vitfil]

solved! 52 seconds...

[@JSSTyger]

Lets see if I got it right. I tried it all in my head and got 423.5.

[@JSSTyger]

Side length AD=14+8. I then used rise over run (8/14) of the diagonal to get DC

[@Teamstudy4595]

1st view

[@PreMath]

Well done, dear Thank you! Cheers! 😀 You are awesome. Keep it up 👍

[@Teamstudy4595]

1st Comment