NonTraditional Method! | Find Area of the Blue Shaded triangle | Important Geometry skills explained

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Learn how to find the area of the Blue Shaded triangle in the right triangle. Important Geometry skills are also explained: area of the triangle formula; similar triangles; Pythagorean theorem. Step-by-step tutorial by PreMath.com
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NonTraditional Method! | Find Area of the Blue Shaded triangle | Important Geometry skills explained
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5 июл 2023

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Комментарии : 47

@wackojacko3962 Год назад

For final solutions , similar triangles have equal ratios ...combined with geometric mean is the ultimate way to go.🙂 I like your non traditional way though.

@PreMath Год назад

Thank you! Cheers! 😀

@abusayeedmaqsood143 Год назад

H0w 9 square is 91?

@PreMath Год назад

@@abusayeedmaqsood143 I made an honest mistake! My apologies...

@davidellis1929 Год назад

Here's a very simple solution. The big triangle ABC has legs 6 and 9, so its legs are in 2:3 ratio. The blue triangle is similar to the big triangle (right angle and shared angle B), so its legs are in the same ratio, say 2x and 3x. By Pythagoras, (2x)^2+(3x)^2=6^2=36, so 13x^2=36 or x=6/sqrt(13). Then the area of the blue triangle is (1/2)(2x)(3x)=3x^2, which is 3*36/13 or 108/13.

@tjwiets6691 Год назад

Did in similar way but used (1:1.5) ratio. Just made it a bit messier under the radical initially. x^2+(1.5x)^2=6^2, 3.25x^2=36, x=sqrt(36/3.25)=sqrt(36x4 / 3.25x4) = 12/sqrt(13). Area = 0.5 * x * 1.5x =108/13

@dawon7750 Год назад

Nice solution as always! But there is a small mistake to the measure of the longest side of the original/biggest triangle. 9x9=81, not 91, in using the pythagorean theorem to find the hypothanuse of the original triangle.

@PreMath Год назад

Thank you! Cheers! 😀

@shanehebert396 Год назад

9^2 = 81 but you got the 117 right :) Once you found the area of ABC, could you then find the length of CD because that area would be the same as using AB as the base so the area would be the same... 1/2 AB * CD = 27. Then you know CD and BC so Pythagorean again to find BD and then area of the triangle BCD = 1/2 BD CD?

@johnfreeman7610 Год назад

👌

@tombufford136 Год назад

Using pythagorous we can calculate AB from sqrt(81+36) .DBC is a similar triangle to ABC so the ratio (sqrt(117): 9: 6) and is the same for DBC with 6, CD, DB. hence sqrt(117)/9 = 6/CD. CD=5. DB=3.32 and area = 0.5 * 3.32 *5= 8.3

@KAvi_YA666 Год назад

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

@murdock5537 Год назад

Nice! Many thanks! This method is really a non traditional approach, Sir, you are great! a = 6; b = 9; c = √(81 + 36) = 3√13 → h = CD = ab/c = 18√13/13 → CD = k → sin⁡(φ) = a/c = 2√13/13 = k/6 → k = 12√13/13 → area ∆BCD = hk/2 = 108/13

@ernestschoenmakers8181 Год назад

One can apply Pythagoras 3 times where you have 3 equations with 3 unknown which can be solved.

@montynorth3009 Год назад

Area of big triangle ABC = 1/2 x 9 x 6 = 27. The 2 inner triangles ADC & DBC are similar. Their corresponding sides, both hypotenuses, are 9 & 6. Their area ratios are proportional to the square of their sides as stated. The ratio therefore is 9^2 / 6^2 or 81 / 36. Thus the white inner triangle has 81 / 117 of the big triangle area ABC. The blue triangle has 36 /117. Blue triangle area = ( 36 / 117) x 27. 8.31.

@soli9mana-soli4953 Год назад

There are many possible solutions. I liked this: CD = 6*cos α (CDB triangle) CD = 9*cos(90°- α) = 9*sin α (ADC triangle) 6*cos α = 9*sin α (dividing by cos α) then we get Tan α =2/3 that means DB/CD = Tan α =2/3 then DB = 2/3*CD so set CD = X by the Pythagorean theorem in the triangle CDB we get: X² + (2/3*X)² = 6² X = 18/√13 Area = X * X*2/3*1/2 Area = (18/√13) * (18/√13)*(2/3)*(1/2) = 108/13

@m.t.v8011 Год назад

Hi premath how you find this type of questions please tell me

@spiderjump Год назад

Let area of triangle CDB be a. Triangles CDB and ADC are similar (a-a) with scale factor of 9/6=3/2 Area of triangle ADC = 9/4a 9/4a + a= 1/2·9·6 13/4a=27 a=27•4/13=108/13

@misterenter-iz7rz Год назад

The area of the large triangle is 6x9/2=27, then the area of the blue triangle is 27x6^2/(6^2+9^2)=27x36/(36+81)=27x36/117=3x36/13=108/13=8.3 approximately. 🙂

@PreMath Год назад

Thank you! Cheers! 😀

@pramodsingh7569 Год назад

Thanks

@PreMath Год назад

You are very welcome! So nice of you. Thank you! Cheers! 😀

@liliyakaloyanova377 Год назад

The triangles ADC and DBC are similar, so CD/DB=9/6=3/2 CD=3.DB/2 Area of CDB A= CD.DB/2 =3.DB^2/4 DB^2=6^2- 9.DB^2/4 13.DB^2=144 A=3.144/4.13

@ybodoN Год назад

Generalization 2D: _the ratio of the areas of two similar figures is equal to the square of the ratio of the corresponding sides._ Generalization 3D: _the ratio of the volumes of two similar solids is equal to the cube of the ratio of the corresponding sides._

@m.t.v8011 Год назад

Super 😊😊😊😊

@alantucker3014 Год назад

You did that a very complicated way. You can just use the ratios of the sides 9/6 = 3 /2 ... ratio of small triangle is 3/2 So sides are 3x and 2x ... (3x)² + (2x)² = 6² ... 13x²=36 ... x²=36/13 Area of triangle = 3x * 2x * 1/2 Area = 3x² ... Area = 108/13

@marioalb9726 Год назад

Hypotenuse: c² = a² + b² c² = 6² + 9² c = 10,8166 cm Area of big triangle: A₁ = b.h/2 = 6 . 9 / 2 = A₁ = 27 cm² Similarity of triangles: Big right triangle is similar of blue right triangle Ratio of areas = (Ratio of any side)² = (6/ 10,8166)² Area of blue right triangle: A₂ = 27 . (6 / 10,8166)² A₂ = 8,307 cm² ( Solved √ )

@honestadministrator Год назад

∆ CDB is similar to ∆ACB [ ∆C D B ] / [ ∆A C B ] = BC^2 / AB^2 = BC^2 / ( AC^2 + BC^2) Hereby [ ∆C D B ] = BC. AC. BC^2 / [2( AC^2 + BC^2)] = 3 x 9 x 4 /( 4 + 9) sqr unit = 108/13 sq unit

@quigonkenny Месяц назад

Triangle ∆BCA: BC² + CA² = AB² 6² + 9² = AB² AB² = 36 + 81 = 117 AB = √117 = 3√13 As ∠BCA = ∠CDB = 90 ° and ∠B is common, triangles ∆BCA and ∆CDB are similar, and ∠BCD = ∠CAB. CD/BC = CA/AB CD/6 = 9/3√13 = 3/√13 √13CD = 18 CD = 18/√13 Blue Triangle ∆CDB: [A] = bcsin(A)/2 = BC•CD(BC/AB)/2 [A] = 6(18/√13)(6/3√13)/2 [A] = 6(18)6/13(6) [A] = 108/13 = 8.31 sq units

@vidyadharjoshi5714 Год назад

Tan A = 6/9 = 0.33. Sin A = CD/AC. CD = 4.98. DB = 3.33 Area = 8.29

@arnejaks5542 Год назад

I HAVE A SIMPLER SOLUTION TO THIS PROBLEM. Let cD =X AND CD^2 =Y SO BY PROP OF SIMILAR TRIANGLES WE HAVE Y = AD×DB Y =SQRROOT{(81=Y)(36-Y) SO Y AND HENCE X CAN BE FOUND AND HENCE BD REQD AREA =X(BD)/2 =108/13

@giuseppemalaguti435 Год назад

A=bh/2=(6cosarctg9/6)*(6sinarctg9/6)/2=108/13

@PreMath Год назад

Thank you! Cheers! 😀

@hollowzangetsu Год назад

This is the method I used too

@himo3485 Год назад

area of the Blue triangle : 6＊9/2 ＊ 6^2/(6^2+9^2) = 27 ＊ 36/117 = 27 ＊ 4/13 = 108/13

@PreMath Год назад

Thank you! Cheers! 😀

@JSSTyger Год назад

I did very rough approximations all the way and get 8.75.

@rishudubey1533 Год назад

😊😊😊❤

@PreMath Год назад

So nice of you. Thank you! Cheers! 😀

@AnonimityAssured Год назад

I worked this out by means of proportions: Spoiler alert. The area of the whole figure is 6 · 9 / 2 = 3 · 9 = 27 square units. The sides of the blue triangle are 2 / 3 those of the white triangle, so the area of the blue triangle is 2² / 3² = 4 / 9 times that of the white triangle, and thus 2² / (3² + 2²) = 4 / (9 + 4) = 4 / 13 times that of the entire figure. The area of the blue triangle is therefore 4 · 27 / 13 = 108 / 13 ≈ 8.3077 square units. Interestingly, if the figure's area is scaled up by a factor of 13 / 9, the result will be an integer, but no successive scaling by the same factor will yield an integer result: 27 · 13 / 9 = 3 · 13 = 39.