MATHCOUNTS Mini #45 -- Maximum Area of Inscribed Rectangles & Triangles

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  • เผยแพร่เมื่อ 23 ธ.ค. 2024

ความคิดเห็น • 43

  • @MrKrabs-xf2tr
    @MrKrabs-xf2tr 4 ปีที่แล้ว +2

    A good formula to always know is that the area of the largest rectangle able to be inscribed within a triangle is always half the area of the triangle.

    • @aayankhare2616
      @aayankhare2616 4 ปีที่แล้ว

      Congratualations you understood the purpose of the video

  • @mateiacd
    @mateiacd 3 ปีที่แล้ว +1

    Excelent idea, thank you ! I think it would have helped if you had assigned letters A,B,C to the vertices of the triangle; D, E, F, G to the vertices of the inscribed rectangle; I, J to the other two points on the base resulted from flipping at 08:40; H to the intersection of those flipped sides

  • @michaelempeigne3519
    @michaelempeigne3519 3 ปีที่แล้ว +2

    Area of triangle using heron's formula = sqrt [ s ( s - a ) ( s - b ) ( s - c ) ] , where s = ( a + b + c ) / 2
    so s = 24 and Area = sqrt [ 24*14*7*3 ] = 84
    84 = ( 1 / 2 ) * 21 * h
    (84 / 21) * 2 = h
    h = 8

  • @Philipel2010
    @Philipel2010 10 ปีที่แล้ว +3

    That was a really clever way to show that the rectangle was the largest possible

  • @richardxu7251
    @richardxu7251 10 ปีที่แล้ว +1

    By algebra we can get the same answer. Say the triangle has length a,b and c with one side of the rectangle with length x on c, and the other side of the rectangle equals d, the height of the triangle on side c equals h, x=c(h-d)/h, S=c(h-d)/h*d/2 =c/h((h-d)h/2). c/h is constant, so S is the biggest when (h-d)h is the biggest.
    (h-d)h=-(d-h/2)^2+1/4*h^2, so the rectangle is the biggest when d=h/2

  • @aashimasinghsisodia7567
    @aashimasinghsisodia7567 4 ปีที่แล้ว +1

    How can we use Heron’s formula in this? I see a lot of comments about this.....

  • @MKWKezer
    @MKWKezer 10 ปีที่แล้ว +3

    Really elegant solution, I love it. :)

  • @howardtang8720
    @howardtang8720 6 ปีที่แล้ว +1

    How do you know that if the side is too short that the top piece will go outside of the triangle?

  • @jwm239
    @jwm239 10 ปีที่แล้ว +1

    ...A splendid demo of a concept of calculus...the thinner and thiner rectanlges, then adjusting their sizes to show why a particular shape can have maximum area. Are MathCounts {tm} students axquainted with Heron's Formula?

    • @anshshah8893
      @anshshah8893 5 ปีที่แล้ว +1

      jwm239 yep if they are decent

  • @ezrasantos9158
    @ezrasantos9158 7 ปีที่แล้ว +3

    You can use Heron's Formula.

    • @ethanli1489
      @ethanli1489 6 ปีที่แล้ว +1

      i thought you said Heroine's Formula I was like WOAH

  • @howardtang8720
    @howardtang8720 6 ปีที่แล้ว +1

    How do you know if one side is smaller that the area will be smaller? Couldn't the other side affect the area as well?

    • @tavishu
      @tavishu 5 ปีที่แล้ว +1

      The other side isn’t increasing as fast as the first is decreasing

  • @BBODANG94
    @BBODANG94 4 ปีที่แล้ว +1

    where is Harvey?????

  • @vnsomeshwar
    @vnsomeshwar 6 ปีที่แล้ว +2

    I used Heron's formula but I accidentally used the full perimeter instead of the semiperimeter in the problem. Whoops! :)

  • @keshavb3128
    @keshavb3128 3 ปีที่แล้ว +1

    I found the other guy finding the area of a rectangle with sides 6.9 and 10 with calculus.
    He used coordinate geometry and got a limit as n→infinity of something and got 69 the long way.

  • @dennisreed7679
    @dennisreed7679 10 ปีที่แล้ว +1

    How would you do this all(go through the though process) in the sprint time period?

    • @Chris-wz9jg
      @Chris-wz9jg 7 ปีที่แล้ว +1

      They probably wouldn't put it in SPRINT.

    • @Chris-wz9jg
      @Chris-wz9jg 7 ปีที่แล้ว +1

      That is HARD.

    • @billyzhao5641
      @billyzhao5641 5 ปีที่แล้ว +1

      @@Chris-wz9jg Not REALLY that hard

    • @billyzhao5641
      @billyzhao5641 5 ปีที่แล้ว +1

      @@Chris-wz9jg They would.I've seen Much harder things than those problems on sprint.

    • @5000subsnovideo
      @5000subsnovideo 4 ปีที่แล้ว +2

      @@Chris-wz9jg
      The proof part should be automatic. The solving part should take like a minute or two at the max.

  • @abdixsimplix2582
    @abdixsimplix2582 5 ปีที่แล้ว +1

    I used an alterative method consisting of a derivative to find maximum.

  • @aakashSky-0
    @aakashSky-0 8 ปีที่แล้ว +1

    can some plz tell me how to derive the formula for Counting Rectangles in an n X m figure.

    • @raghavnarula1057
      @raghavnarula1057 8 ปีที่แล้ว +2

      Assuming a = the number of horizontal lines and b = the number of vertical lines, then the formula is aC2 * bC2 (C means combination). In a n x m grid, that would have n+1 vertical lines, and m+1 horizontal lines, so the number of rectangles would be ((n+1)C2) * ((m+1)C2). Basically, you are picking two horizontal lines for sides of the rectangles and two vertical lines to be the other two sides of the rectangle. Note: This formula works best for grids, but if dots/points are given instead, in addition to this formula, you have to count all the diagonal rectangles.

  • @richardxu7251
    @richardxu7251 10 ปีที่แล้ว +1

    What happened in the end? The last sentence after the solution.

    • @edgarwang4659
      @edgarwang4659 9 ปีที่แล้ว +1

      Richard Xu inside joke about richard's imaginary friend named harvey who "does his geometry for him".

  • @lemniscate9718
    @lemniscate9718 5 ปีที่แล้ว +1

    Dude...i loved it

  • @wiktorl541
    @wiktorl541 7 ปีที่แล้ว +2

    use heron's formula

  • @richardxu7251
    @richardxu7251 10 ปีที่แล้ว +1

    If the triangle is not obtuse, the three cases will result in the same answer, area of the triangle/2.

  • @keshavb3128
    @keshavb3128 3 ปีที่แล้ว +1

    there's a much faster method to find area of triangle, a rectangle minus the 6-8-10 and 5-8-17 triangles.

  • @Deathranger999
    @Deathranger999 10 ปีที่แล้ว +1

    Or you could've just used Heron's Formula. :P A whole lot easier IMO.

    • @pi_is_3.147
      @pi_is_3.147 3 ปีที่แล้ว +1

      yea, would be alot easier lol

  • @_mstang
    @_mstang 10 ปีที่แล้ว +1

    USAMTS shirt! :)

  • @StardustSifr1
    @StardustSifr1 9 ปีที่แล้ว +2

    heron

  • @Phurngirathaana
    @Phurngirathaana 5 ปีที่แล้ว +1

    42,the answer's always 42

    • @5000subsnovideo
      @5000subsnovideo 4 ปีที่แล้ว +2

      I see, you are a man of culture as well.

  • @nsggamer8554
    @nsggamer8554 4 ปีที่แล้ว +1

    Captain America