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

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  • Опубликовано: 24 дек 2024

Комментарии • 32

  • @patrasujata3627
    @patrasujata3627 Год назад +2

    Great solution sir

    • @PreMath
      @PreMath  Год назад +2

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

    • @patrasujata3627
      @patrasujata3627 Год назад

      Welcome sir

  • @parthtomar6987
    @parthtomar6987 Год назад +2

    Nice solution

    • @PreMath
      @PreMath  Год назад +1

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

  • @williamwingo4740
    @williamwingo4740 Год назад +5

    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. 🤠

  • @KAvi_YA666
    @KAvi_YA666 Год назад

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

  • @nina_alb
    @nina_alb Год назад

    Nice solution:)

  • @abstragic4216
    @abstragic4216 Год назад +1

    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
    @robertlynch7520 8 месяцев назад

    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
    @ybodoN Год назад +1

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

  • @rey-dq3nx
    @rey-dq3nx 3 месяца назад

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

  • @arnavkange1487
    @arnavkange1487 Год назад +2

    I respect u sir and also like your sums

    • @PreMath
      @PreMath  Год назад

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

  • @davidthomas6043
    @davidthomas6043 Год назад +1

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

    • @PreMath
      @PreMath  Год назад

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

  • @marioalb9726
    @marioalb9726 Год назад +1

    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 √ )

  • @soli9mana-soli4953
    @soli9mana-soli4953 Год назад

    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
      @ybodoN Год назад +1

      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
      @soli9mana-soli4953 Год назад

      Very good! Congratulation!

  • @bigm383
    @bigm383 Год назад +1

    Thanks Professor. Most excellent!❤

    • @PreMath
      @PreMath  Год назад

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

  • @Copernicusfreud
    @Copernicusfreud Год назад

    Yay! I solved the problem.

  • @fadetoblah2883
    @fadetoblah2883 Год назад

    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.

  • @teambellavsteamalice
    @teambellavsteamalice Год назад +1

    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).

  • @unknownidentity2846
    @unknownidentity2846 Год назад +2

    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
      @PreMath  Год назад

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

  • @dahibhalaa
    @dahibhalaa Год назад

    Ab + dc

  • @gelbkehlchen
    @gelbkehlchen Год назад

    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

  • @devondevon4366
    @devondevon4366 Год назад

    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.

  • @devondevon4366
    @devondevon4366 Год назад

    11.6

  • @JSSTyger
    @JSSTyger Год назад

    11.56 I say