Can you find area of the Green shaded region? | (Triangle) |

Learn how to find the area of the Green shaded region in the triangle. Important Geometry skills are also explained: Pythagorean Theorem; area of triangle formula. Step-by-step tutorial by PreMath.com
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Can you find area of the Green shaded region? | (Triangle) | #math #maths | #geometry
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[@luigipirandello5919]

Nice problem. Thank you Sir.

[@PreMath]

I'm glad you enjoyed it!😀 You are very welcome! Thanks for the feedback ❤️

[@nandisaand5287]

Pretty straightforward problem. Thanks for the quick morning brain calisthenics, Prof.

[@uwelinzbauer3973]

As I saw the 13 in the Hypothenuse, I immediately had a certain suspect: the 5:12:13 primitive pythagorean integer number triple... After the calculations that suspect was confirmed! Stretched by the factor 2 through the board. Nice geometry question, fun to solve 😀, thanks and a happy Sunday 😊

[@prossvay8744]

(9x+6)^2+(2x+6)^2=(13x)^2 So x=2 Green area=1/2(18+6)(4+6)-1/2(4)(18)=84 square units.❤❤❤

[@santiagoarosam430]

(6+9x)²+(6+2x)²=(13x)²---> x=2---> AEDC=AEC+EDC =6*10/2 +6*18/2 =30+54 =84. Gracias y saludos

[@marioalb9726]

(13x)² = (9x+6)² + (2x+6)² 13²x²=(9²x²+108x+36)+(4x²+24x+36) 169x²-81x²-4x²=108x+24x+36+36 84x²-132x-72=0 x = 2 cm A₁ = ½ (9x+6)(2x+6) = 120cm² A₂ = ½ 9x 2x = ½*18*4 = 36cm² Green shaded area: A = A₁-A₂ = 120-36 = 84 cm² ( Solved √)

[@ناصريناصر-س4ب]

According to the Pythagorean theorem, we have (6+2x)²+(6+9x)²=(13x)², from which x=2. We have that the area of triangle ABC is (6+2x)(6+9x)/2=120, and the area of triangle BDE is 2x*9x/2=36. From which the area of quadrilateral ACDE is 120-36=84

[@شرکتراهوساختمانیقرهقشون]

84

[@alster724]

Easy

[@RealQinnMalloryu4]

(6)^2=36 +{13x+13x ➖ }=26x^2 +{9x+9x ➖}=18x^2 +{2x+2x ➖ }=4x^2 {26x^2+18x^2+4x^2}=48x^6,/36=1.12x^6 1.2^6x^6 1.2^2^3x2^3 1.1^2^1x^1^3 2x^3 (x ➖ 3x+2).

[@marcgriselhubert3915]

Pythagorean theorem in ABC: (13.x)^2 = (9.x +6)^2 + (2.x +6)^2. When simplified we obtain:7.(x^2) - 11.x -6 = 0. Delta = 289 = 17^2, so x = 2 or x = -3/7 (rejected). So x = 2. the area of aBC is (24.10)/2 = 120, the area of EBD = (4.18)/2 = 36, so the green area is 120 - 36 = 84. (Very easy.)

[@Birol731]

My way of solution is ▶ For the right triangle ΔABC [AB]= 6+9x [BC]= 6+2x [CA]= 13x By applying the Pythagorean theorem, we can write: [AB]²+[BC]²= [CA]² (6+9x)²+(6+2x)²= (13x)² 36+108x+81x²+36+24x+4x²= 169x² Combining like terms gives: 169x²-4x²-81x²= 72+132x 84x² - 132x-72=0 Dividing both sides by 12: 7x²-11x-6=0 Calculating the discriminant: Δ= 121-4*7*(-6) Δ= 289 √Δ= 17 ⇒ x₁= (11+17)/14 x₁= 2 x₂= (11-17)/14 x₂= - 3/7 ❌ x₂ < 0 ⇒ x= 2 [AB]= 6+9x [AB]= 24 [BC]= 6+2x [BC]= 10 A(ΔABC)= 24*10/2 A(ΔABC)= 120 square units A(ΔEBD)= 9x*2x/2 A(ΔEBD)= 9x² A(ΔEBD)= 36 square units Agreen= A(ΔABC) - A(ΔEBD) Agreen= 120 - 36 Agreen= 84 square units ✅

[@misterenter-iz7rz]

(2x+6)^2+(9x+6)^2=(13x)^2, 169x^2=85x^2+132x+72, 84x^2-132x-72=0, 7x^2-11x-6=0, (7x+3)(x-2)=0, x=2, -3/7 rejected.then the area can be gotten spontaneously. 😅

[@quigonkenny]

Triangle ∆ABC: AB² + BC² = CA² (9x+6)² + (2x+6)² = (13x)² 81x² + 108x + 36 + 4x² + 24x + 36 = 169x² 84x² - 132x - 72 = 0 7x² - 11x - 6 = 0 7x² - 14x + 3x - 6 = 0 7x(x-2) + 3(x-2) = 0 (x-2)(7x+3) = 0 x = 2 | -x = -3/7- ❌ x ≥ 0 The green area will be the area of ∆ABC minus the area of ∆EBD. Green quadrilateral AEDC: A = AB(BC)/2 - EB(BD)/2 A = (6+9(2))(6+2(2))/2 - 9(2)(2(2))/2 A = (6+18)(6+4)/2 - 18(4)/2 A = (24(10)-72)/2 A = (240-72)/2 = 168/2 A = 84 sq units

[@adgf1x]

green shaded area=120-36=84sq unit.ans

[@alexundre8745]

Bom dia Mestre Essa Acertei Graças a Deus e ao Sr estou aprendendo Geometria Deus Lhe Abençoe Shalom Baruch Adonai 🇧🇷🇮🇱🇺🇸

[@wasimahmad-t6c]

X=2)(24×10=240÷2=140-18×2=84

[@KeithAllen-pg8ep]

The 13x hints at the possibility of a multiple of a (5,12,13) triangle, so try x = 2 and we have a (10,24,26) triangle - BINGO! It's plain sailing from there.

[@gelbkehlchen]

Solution: Pythagoras: AB²+BC² = AC² ⟹ (6+9x)²+(2x+6)² = (13x)² ⟹ 36+108x+81x²+4x²+24x+36 = 169x² ⟹ 132x+85x²+72 = 169x² |-85x²-132x-72 ⟹ 84x²-72-132x = 0 |/84 ⟹ x²-11/7*x-6/7 = 0 |p-q-formula ⟹ x1/2 = 11/14±√[(11/14)²+6/7] = 11/14±17/14 x1 = 11/14+17/14 = 28/14 = 2 and x2 = 11/14-17/14 = -6/14 = -3/7 [not guilty in geometry] ⟹ green area = AB*BC/2-EB*BD/2 = (6+9*2)*(2*2+6)/2-9*2*2*2/2 = 12*10-36 = 84

[@sergioaiex3966]

Solution: Applying Pythagorean Theorem in triangle ABC, we have: (6 + 9x)² + (2x + 6)² = (13x)² 36 + 108x + 81x² + 4x² + 24x + 36 = 169x² 84x² - 132x - 72 = 0 (÷12) 7x² - 11x - 6 = 0 ∆ = (-11)² - 4 . 7 . -6 ∆ = 121 + 168 ∆ = 289 x = - (-11) ± √(289)/2 . 7 x = 11 ± 17/14 x' = 2 Accepted x" = -6/14 = - 3/7 Rejected Green Area = A ABC - A EBD Green Area = ½ 24 . 10 - ½ 18 . 4 Green Area = 120 - 36 Green Area = 84 Square Units ✅

[@unknownidentity2846]

Let's find the area: . .. ... .... ..... The triangle ABC is a right triangle, so we can apply the Pythagorean theorem: AC² = AB² + BC² AC² = (AE + BE)² + (BD + CD)² (13x)² = (6 + 9x)² + (2x + 6)² 169x² = 36 + 108x + 81x² + 4x² + 24x + 36 84x² − 132x − 72 = 0 7x² − 11x − 6 = 0 x = {11 ± √[11² − 4*7*(−6)]}/(2*7) = [11 ± √(121 + 168)]/14 = (11 ± √289)/14 = (11 ± 17)/14 Since x>0, we have only one useful solution: x = (11 + 17)/14 = 28/14 = 2 Now we are able to calculate the area of the green region: AB = AE + BE = 6 + 9x = 6 + 9*2 = 6 + 18 = 24 BC = BD + CD = 2x + 6 = 2*2 + 6 = 4 + 6 = 10 BD = 2x = 2*2 = 4 BE = 9x = 9*2 = 18 A(green) = A(ABC) − A(BDE) = (1/2)*AB*BC − (1/2)*BD*BE = (1/2)*24*10 − (1/2)*4*18 = 120 − 36 = 84 Best regards from Germany

[@JSSTyger]

A = 18+38(5+√67)/7

[@JSSTyger]

One little slip up let to this. Somehow my brain said (6+2x)² = 36+12x+4x²

[@himo3485]

(6+9x)²+(6+2x)²=(13x)² 36+108x+81x²+36+24x+4x²=169x² 84x²-132x-72=0 7x²-11x-6=0 (7x+3)(x-2)=0 x>0 , x=2 AB=6+9x=24 BC=6+2x=10 CA=13x=26 EB=9x=18 BD=2x=4 area of Green shaded region : 24*10*1/2 - 18*4*1/2 = 120 - 36 = 84

[@cyruschang1904]

(6 + 2x)^2 + (6 + 9x)^2 = (13x)^2 85x^2 + 132x + 72 = 169x^2 84x^2 - 132x - 72 = 0 21x^2 - 33x - 18 = 0 7x^2 - 11x - 6 = 0 (x - 2)(7x + 3) = 0, x = 2 white area = (2x)(9x)/2 = 9x^2 Large triangle area = (6 + 2x)(6 + 9x)/2 = (3 + x)(6 + 9x) = 9x^2 + 33x + 18 Green area = (9x^2 + 33x + 18) - 9x^2 = 33x + 18 = 66 + 18 = 84

[@sorourhashemi3249]

Thanks. Easy

[@cyruschang1904]

@sorourhashemi3249 A lot of simplification steps though 🙂

[@sorourhashemi3249]

@cyruschang1904 thanks for sending wonderful geometrical tests by which I learned so much from them. There are so many challenging and difficult ones which I couldn't solve them either. So, looking at your solving ways, I've learned so much from you. Thanks again and good luck🥰

[@cyruschang1904]

@@sorourhashemi3249 Glad you find it useful 🙂

[@LuisdeBritoCamacho]

STEP-BY-STEP RESOLUTION PROPOSAL : 01) Area of Green Shaded Region (GSRA) = (6 + 2X) * (6 + 9X) - (2X * 9X) /2 ; GSRA = (36 + 54X + 12X + 18X^2 - 18X^2 / 2 ; GSRA = 36 + 66X) / 2 ; GSRA = (18 + 33X) sq cm 02) (6 + 2X)^2 + (6 + 9X)^2 = (13X)^2 03) Solutions : X = - 3/7 or X = 2 04) GSRA = 18 + 33(2) ; GSRA = 18 + 66 ; GSRA = 84 sq cm OUR BEST ANSWER : The Area of Green Shaded Region equal to 84 Square Units.

[@misterenter-iz7rz]

Plenty of computation 😢

[@AndreasPfizenmaier-y7w]

Not too difficult