Can you find the area of the Yellow Square? | (Important Geometry skills explained) |

Learn how to find the area of the Yellow square inscribed in the right triangle. Important Geometry skills are also explained: area of the square formula; Pythagorean Theorem; similar triangles. Step-by-step tutorial by PreMath.com
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[@patrasujata3627]

Great solution sir

[@PreMath]

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

[@patrasujata3627]

Welcome sir

[@williamwingo4740]

As usual, a somewhat alternative approach: The big triangle is integer Pythagorean, 5-12-13. Because everything is perpendicular, all three smaller triangles are similar to each other and to the big one by angle-angle-angle. Using your notation, consider triangles ADE and EBF and compare each to the big triangle ABC: From triangle ADE, L/x = 13/12; cross-multiply and 13x = 12L; so x = 12L/13. From triangle EBF, (12 -- x)/L = 13/5; cross-multiply and we get 60 -- 5x = 13L; so x = (60 -- 13L)/5. We now have two independent expressions for x in terms of L. equating them, we get: x = 12L/13 = (60 -- 13L)/5; cross-multiply to get 60L = (13)(60 -- 13L) = 780 -- 169L. Solve for L: 60L + 169L = 229L = 780; so L = 780/229, just as you determined; and the area of the square is (780/229)^2. We could use any two of the smaller triangles and it would work out the same way. (I almost said "similarly" but wouldn't want to make a bad pun.) Cheers. 🤠

[@parthtomar6987]

Nice solution

[@PreMath]

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

[@abstragic4216]

Length BC = the sum of lengths CP, PF and FB. By Pythagorous BC = 13. Using similar triangles CP = 5L/12 and FB = 12L/5. Therefore 5L/12 + L + 12L/5 = 13. Using a common denominator of 60 on the LHS expression, (60L+25L+144L)/60 =13 which simplifies to L = 780/229.

[@robertlynch7520]

Perhaps you have figured out these kind of problems "in your head?" I decided to focus on the [5] side of the triangle. first though, calculating the hypotenuse of [13] was helpful. I used 𝒔 as the side of the yellow square, "𝒔" for "Square" [1.1]  △CPD … DP is on the '12' leg. [1.2]  △ADE … DE is on the '13' leg. Put 'em together: [2.1]  13(𝒔/12) ⊕ 5(𝒔/13) = 5 The rationale is "divide by the normalized side-length, then multiply by the side-length that we need in proportion". Find a common denominator … [3.1]  (13 / 13)13(𝒔/12) + (12 / 12)5(𝒔/13) = ((12 × 13) / (12 × 13)) × 5 Then eliminate the denominator [4.1]  169𝒔 + 60𝒔 = 780 [4.2]  229𝒔 = 780 [4.3]  𝒔 = 780 ÷ 229 and the yellow square [5.1]  𝒔² = 3.406² = 11.602 Ta, da!

[@ybodoN]

Generalized: s = abc / (ab + c²) where s = DE (side of the square), a = CA, b = AB and c = BC

[@marioalb9726]

C² = 5²+12² C = 13 cm L / cos α + L sin α = 5 13 L /12 + 5 L 13 = 5 L (13/12 + 5/13) = 5 1,4679 L = 5 L = 3,406 cm Area = L² Area = 11.6 cm² ( Solved √ )

[@rey-dq3nx]

5x/12+x+12x/5=13 (144x+25x+60x)/60=13 229x=780 Area=11.6 sq units

[@teambellavsteamalice]

Darn, almost had it by calculating in my head. Made one error though. 😞 I only used one variable, the one for L. AE = 12/13*L and EB = 13/5*L So L = 12 / (12/13 + 13/5) = 12 * 5 * 13 / ( 12*5 + 13*13) here I made the error and used 12*13=156 instead of 13*13, so 216 instead of 229. Too bad... So L^2 = (780 / 229)^2 CD + DA = 5 would have been a similar route. 13/12*L + 5/13*L = 5 with again L = 5*12*13/(13*13+5*12).

[@davidthomas6043]

Very interesting. The various equations to get the sides are really a version of trigonometry.

[@PreMath]

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

[@unknownidentity2846]

Pythagorean theorem (right triangle ABC): BC² = AB² + AC² = 12² + 5² = 144 + 25 = 169 ⇒ BC = 13 Similar triangles: ADE: AD:AE:DE = 5:12:13 CDP: CP:DP:CD = 5:12:13 AD/DE = 5/13 DP/CD = 12/13 DE = DP ⇒ (AD/DE)*(DP/CD) = AD/CD = (5/13)*(12/13) = 60/169 AD + CD = AC = 5 AD = (60/169)CD (60/169)CD + CD = 5 (229/169)CD = 5 CD = 5*169/229 DP = (12/13)*CD = (12/13)*5*169/229 = 780/229 A(square) = DP² = (780/229)² Best regards from Germany

[@PreMath]

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

[@KAvi_YA666]

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

[@bigm383]

Thanks Professor. Most excellent!❤

[@PreMath]

You're very welcome! Thanks for your feedback! Cheers! 😀 Kind regards 👍

[@nina_alb]

Nice solution:)

[@arnavkange1487]

I respect u sir and also like your sums

[@PreMath]

Thank you, dear! Cheers! 😀 Thanks for your feedback! Cheers! 😀 You are the best❤️ . Keep it up 👍

[@devondevon4366]

Answer 11.6016 or 11.6 I did something different to see if it works First, all four triangles are similar since at least two angles of each are congruent Let the side of the square = 780 b. I used 780 b (13*12*5= 780) in order to avoid decimals and to make it easier to calculate the other sides since these triangles are similar Hence area of the square = 780b * 780 b Hence CP = 325b (5/12 of 780) since CP is the '5' of triangle CPB CD =845 b (13/12 of 780) Recall I let the length of the square = 780 b DA= 300 b (5/13 of 780) AE = 720 b (12/13 *780 ) EB= 2028 b (13/5 * 780 ) as EF is the '5' here and BE is the '13' FB =1872 b (12/5 * 780) EF again is the '5' here but is the "12" Hence AC=CD+ DA = 1145b AB= AE + EB = 2748 b So the height and base of the square are 1145 b and 2748 b Hence the area of the triangle in terms of b =3,146,460/2 = 1,573,230 b^2 And the area of the square in terms of b = 608,400 b^2 (780b * 780b) Hence the square as a percentage of the triangle = 608, 400/1,573, 230 = 0.38672 or 38.672% Since the area of the triangle = 30 [ 5 * 12 * 1/2], then The area of the square is 0.38672 * 30 = 11.6016 Answer The square is approximately 38.67% of the main triangle.

[@misterenter-iz7rz]

Let s be the side of the square, CD=13s/12, DA=5s/13, thus 5=(13/12+5/13)s=(13^2+60)s/13x12=229s/156, s=156x5/229=780/229, therefore the area is 780^2/229^2=11.6 approximately. 😅

[@PreMath]

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

[@fadetoblah2883]

The numbers made the final calculations a little unwieldy to perform with pen and paper alone (at least for me) so I stopped after finding L = 780/229 and convincing myself that this fraction could not be simplified. An excellent problem apart from that. Thanks.

[@soli9mana-soli4953]

Once known that ABC is the Pythagorean triplet (5,12,13) we can easily show that all the right triangles inside ABC are similar to ABC, so I wrote this system: 12x+13z=12 13y+5x=5 5y+12y+12z=13 (x for DAE, y for DCP and z for EFB) and found y = 845/2977 = 65/229 = 0,2838... so side of square = 12y = 12*0,28= 3,406... area = 11,60...

[@ybodoN]

If you start with 13x = 12y = 5z (i.e. the side of the square), 5x + 13y = 5 or 12x + 13z = 12 is enough to solve for x, y or z. For example: 13x = 12y ⇒ y = 13x / 12 so 5x + 13y = 5 becomes 5x + 13 (13x / 12) = 5 and therefore x = 5 / (5 + 13² / 12)

[@soli9mana-soli4953]

Very good! Congratulation!

[@Copernicusfreud]

Yay! I solved the problem.

[@gelbkehlchen]

Solution: BC = √(5²+12²) = 13. There are 4 similar right triangles, ABC, DPC, AED and EBF. The side x of the square is in triangle AED hypotenuse, in triangle DPC long leg and in triangle EBF short leg. AD = a. Similarity triangle DPC to triangle ABC: (1) x/(5-a) = 12/13 Similarity triangle AED to triangle ABC: (2) a/x = 5/13 |*x ⟹ (2a) a = 5/13*x |in (1) ⟹ (1a) x/(5-5/13*x) = 12/13 |*(5-5/13*x) ⟹ (1b) x = 12/13*(5-5/13*x) = 60/13-60/169*x |+60/169*x ⟹ (1c) x+60/169*x = 229/169*x = 60/13 |*169/229 ⟹ (1d) x = 60/13*169/229 = 60*13/229 = 780/229 ⟹ area of the yellow square = x² = (780/229)² ≈ 11,6016

[@dahibhalaa]

Ab + dc

[@devondevon4366]

11.6

[@JSSTyger]

11.56 I say