Can you find area of the Blue Square? | (Justify your answer) |

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  • เผยแพร่เมื่อ 24 ม.ค. 2025

ความคิดเห็น • 93

  • @zupitoxyt
    @zupitoxyt 15 วันที่ผ่านมา +5

    Here's another exciting and challenging question ahead.
    Thanks professor

    • @PreMath
      @PreMath  15 วันที่ผ่านมา +2

      Glad you enjoyed the challenge! 👍
      You are very welcome!
      Thanks for the feedback ❤️🙏

    • @zupitoxyt
      @zupitoxyt 15 วันที่ผ่านมา +2

      @PreMath I welcome you ☺️

  • @harikatragadda
    @harikatragadda 15 วันที่ผ่านมา +20

    By Similarity of ∆CPB and ∆MDC, PC = 4/2=2 and BC² = 20= Area

    • @egillandersson1780
      @egillandersson1780 15 วันที่ผ่านมา +1

      Same for me : solvable this way in less then 20 sec

    • @PreMath
      @PreMath  14 วันที่ผ่านมา +1

      Excellent!
      Thanks for sharing ❤️

    • @T0NYD1CK
      @T0NYD1CK 14 วันที่ผ่านมา

      @@egillandersson1780 Same for me - except that it took much longer than 20 seconds! I must be getting slow in my old age. It took me a long time to realise that the side of the square was equal to twice the length of half of it!

  • @stephendavis6408
    @stephendavis6408 15 วันที่ผ่านมา +2

    Thanks!

    • @PreMath
      @PreMath  15 วันที่ผ่านมา

      You are very welcome!
      Thank you very much for supporting my youtube channel of the work I do.
      Kind regards🙏❤️

  • @andrepiotrowski5668
    @andrepiotrowski5668 15 วันที่ผ่านมา +16

    Way too complicated! MDC and BPC are similar, the ratio of the catheti is 1:2, so PC is 2. 4²+2² = 20 = (BC)² , which is the area we were searching.

    • @URMBOT
      @URMBOT 15 วันที่ผ่านมา +5

      So much better! 😊

  • @georgebliss964
    @georgebliss964 15 วันที่ผ่านมา +4

    Similar triangles DMC & PCB.
    X / 2X = PC / 4.
    PC = 4X / 2X = 2.
    CB^2 = 4^2 + PC^2.
    CB^2 = 16 + 4 = 20.
    CB^2 = area of square.

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for sharing ❤️

  • @johnwindisch1956
    @johnwindisch1956 12 วันที่ผ่านมา

    Thanks to you sir!
    And to the other solutions offered in the posts

  • @d-hat-vr2002
    @d-hat-vr2002 14 วันที่ผ่านมา

    I had one of those rare 'outside the box' moments with this question, where the Universe delivered an insight to my dull mathematical mind.
    Up to 3:11 where Mr. Premath finds MC=x√5 my process was the same. But then I extended CM and BA, to intersect off to the left at a point K.
    By showing △CDM ≅ △KAM by ASA, we see that area □ABCD = area △BCK
    Then calculating the area of △BCK two different ways and setting them equal reveals the value of x, as follows:
    (½)(2⋅CM)(PB) = (½)(2⋅AB)(BC)
    (½)(2x√5)(4) = (½)(4x)(2x)
    etc.

  • @FSIec-x3x
    @FSIec-x3x 15 วันที่ผ่านมา +3

    Use similar triangles to figure out CM=4+4/(2^2)=5 and Blue Square Area BP×CM=4×5=20

    • @jamesgodfrey8400
      @jamesgodfrey8400 15 วันที่ผ่านมา

      Yep - I did that in my head in about 30s

    • @dan-florinchereches4892
      @dan-florinchereches4892 15 วันที่ผ่านมา

      I think if we denote the side length of square as X by similar triangles we get
      X/4=cm/X
      From Pythagoras theorem cm=sqrt((X/2)^2+x^2)=X*√5/2
      => X/4=√5/2
      X=2√5 so area is 20 you guys are totally right

    • @FSIec-x3x
      @FSIec-x3x 14 วันที่ผ่านมา

      It's not necessary to consider the side of the square,
      SquareArea=2×ΔBCMarea=BP×CM

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for sharing ❤️

  • @MrPaulc222
    @MrPaulc222 15 วันที่ผ่านมา +2

    Call the square's sides 2x. The area will be 4x^2
    BPC is similar to MDC.
    (2x)/4 = (MC)/(2x)
    4*(MC) = 4x^2
    Therefore, MC = x^2
    Sides are now:
    x. 2x, x^2
    (PC), 4, 2x
    (2x)/(x) = (4)/(PC)
    4x = (2x)*(PC), so PC = 2.
    Ref BPC: 4^2 + 2^2 = 4x^2
    16 + 4 = 4x^2
    20 = 4x^2 which is the square's area.
    I used 2x as the square's sides rather than plain x. This was becaise there was an early warning that half sides were involved and I wanted to avoid too many fractions.

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for sharing ❤️

  • @jamestalbott4499
    @jamestalbott4499 15 วันที่ผ่านมา +1

    Thank you!

    • @PreMath
      @PreMath  15 วันที่ผ่านมา

      Thanks for watching! ❤️
      You are very welcome!

  • @joseeduardomachado3436
    @joseeduardomachado3436 15 วันที่ผ่านมา +1

    Exercício interessante

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Glad to hear that!
      Thanks for the feedback ❤️🙏

  • @OsazGeorge
    @OsazGeorge 12 วันที่ผ่านมา

    Using similar triangle approaches, we already know that PC should be half of 4 = 2. Since triangle CDM shows that DM = x and CD = 2x.
    The area of the square with length 2x is 4x^2.
    From Pythagoras, 4x^2 = 4^2 + 2^2 =20 (QED)

  • @alster724
    @alster724 15 วันที่ผ่านมา +1

    There is a faster way to solve this.
    Since we know MC= x√5 we can immediately substitute it in the Similarity Formula to get x= √5
    (x√5)/2x= 2x/4
    √5/2= x/2
    x= √5
    Since we know that the square's area is 4x²
    4(√5)²
    4(5)
    A= 20 units²

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for the feedback ❤️

  • @marioalb9726
    @marioalb9726 15 วันที่ผ่านมา +1

    Similarity of right triangles:
    4/s = s / √[s²+(½s)²]
    s² = 4 √[5/4 s²] = 4 √5/2 s
    s² = (2√5)² = 20 cm² ( Solved √ )

  • @yakovspivak962
    @yakovspivak962 11 วันที่ผ่านมา

    Let 2X - side of the square,
    Then, for similar triangles:
    2X / X = 4 / ((4X^2-16)^.5)
    X = 5^0.5, S = 20

  • @imetroangola17
    @imetroangola17 15 วันที่ผ่านมา +2

    *Another solution:*
    Draw the segment BM. The triangle ∆BMC is isosceles, since BM = MC.Let h be the height drawn from vertex M in ∆BMC, then we can find the area of this triangle as follows:
    h × BC/2 = BP×MC/2
    *h × BC= 4MC (1).*
    By Pythagoras in ∆GCD, we have:
    MC² = MD² + DC² = (DC/2)² + DC²
    MC² = DC²(1/4 + 1) = DC²/4 × 5
    MC = √5DC/2, being DC = BC, then MC = √5BC/2. Substituting in (1):
    h × BC= 4 √5BC/2
    h = 2√5 = √20. Now, h= AB=DC=AD=BC. Thus, the area of the square is given by h², i.e,(√20)² = *20.*

    • @PreMath
      @PreMath  14 วันที่ผ่านมา +1

      Excellent!
      Thanks for sharing ❤️

  • @Waldlaeufer70
    @Waldlaeufer70 15 วันที่ผ่านมา +1

    s^2 = 4^2 + 2^2
    s^2 = square area = 16 + 4 = 20 square units

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for sharing ❤️

  • @marioalb9726
    @marioalb9726 15 วันที่ผ่านมา +2

    A = s² = (4/cos[atan(1/2)] )²
    A = 20 cm² ( Solved √ )

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for sharing ❤️

  • @shashijoshi8039
    @shashijoshi8039 15 วันที่ผ่านมา

    A very good solution.
    Another alternate approach will be:
    After determining that MC = x. Sqrt5,
    Join points M and B.
    Then the area of triangle MBC can be calculated in 2 ways as
    1/2.MC.4 , and 1/2.2x.2x
    Equating these 2 values, we will get 4.MC = 4.xsq = area of the square ABCD!

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Glad to hear that!
      Thanks for the feedback ❤️

  • @unknownidentity2846
    @unknownidentity2846 15 วันที่ผ่านมา +3

    Let's find the area:
    .
    ..
    ...
    ....
    .....
    First of all we apply the Pythagorean theorem to the right triangle CDM. With s being the side length of the square we obtain:
    CM² = CD² + DM² = CD² + (AD/2)² = s² + (s/2)² = s² + s²/4 = 4s²/4 + s²/4 = 5s²/4 ⇒ CM = √(5s²/4) = (√5/2)s
    Since ∠BPC=∠CDM=90° and ∠CBP=∠DCM, we know that the triangles BCP and CDM are similar. So we can conclude:
    BC/BP = CM/CD
    s/4 = (√5/2)s/s
    s/4 = √5/2
    ⇒ s = 4*√5/2 = 2√5
    Now we are able to calculate the area of the blue square:
    A(ABCD) = s² = (2√5)² = 20
    Best regards from Germany

    • @zupitoxyt
      @zupitoxyt 15 วันที่ผ่านมา

      Still keen eye on you 🤨

    • @PreMath
      @PreMath  14 วันที่ผ่านมา +1

      Excellent!😀
      Thanks for sharing ❤️

  • @zawatsky
    @zawatsky 15 วันที่ผ่านมา

    Разрезаем по линии МС, поворачиваем ▲CDM на 180° вокруг точки М, складываем MD с АМ. В результате из малого треугольника и трапеции получается большой треугольник, подобный малому (вдвое больше), с площадью, равной площади квадрата. РВ является его высотой. Точку, в которую перешла т. С, обозначим за Е. ▲ВСЕ прямоугольный, потому площадь также равна произведению катетов. Сторону квадрата за х, тогда х*2х=2х²=4*СЕ/2=2СЕ⇔х²=СЕ. Но СЕ²=х²+(2х)², как гипотенуза, получаем уравнение четвёртой степени: х⁴=х²+4х²=5х². Отрицательные и комплексные корни искать не нужно, потому смело сокращаем: х²=5, откуда х=√5. Можно решать и через пропорцию, заметив, что катеты 1 к 2, гипотенуза кратна √5, а высота, соответственно, должна быть кратна 1/√5. 4х/√5=4, откуда х=√5.

    • @PreMath
      @PreMath  14 วันที่ผ่านมา +1

      Thanks for sharing ❤️

  • @giuseppemalaguti435
    @giuseppemalaguti435 15 วันที่ผ่านมา +1

    √(5l^2/4-16)+√(l^2-16)=√5l/2...l^2=20

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for sharing ❤️

  • @marioalb9726
    @marioalb9726 15 วันที่ผ่านมา +2

    A = 4 A₁ + A₂ = 4 (½b.h) + (s₂)²
    A = 4 (½*4*2) + (½4)² = 16 + 4
    A = 20 cm² ( Solved √ )

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for sharing ❤️

  • @AmirgabYT2185
    @AmirgabYT2185 15 วันที่ผ่านมา +2

    S=20

    • @PreMath
      @PreMath  14 วันที่ผ่านมา +1

      Excellent!
      Thanks for sharing ❤️

  • @imetroangola17
    @imetroangola17 15 วันที่ผ่านมา +2

    *Solution:*
    In ∆MCD:
    Let ∠DCM = α. Hence,
    tg α= MD/DC = MD/2MD= 1/2.
    In ∆BCP:
    tg (90 - α) = 4/PC →1/tg α = 4/PC
    2 = 4/PC → PC = 2. By Pythagoras, we have:
    BC² = 4² + 2² = 20, this tells us that the area of the square is 20.

    • @PreMath
      @PreMath  14 วันที่ผ่านมา +1

      Excellent!
      Thanks for sharing ❤️

  • @mcgraphix
    @mcgraphix 13 วันที่ผ่านมา

    If x^2 = x√5, why not just divide both sides by x to get √5?

  • @cyruschang1904
    @cyruschang1904 15 วันที่ผ่านมา +1

    We have two similar right triangles
    If the square side length = 2x
    (√5)x/2x = 2x/4
    4x^2 = 4(√5)x
    2x = 2√5
    Square area = (2√5)^2 = 20

    • @PreMath
      @PreMath  14 วันที่ผ่านมา +1

      Excellent!
      Thanks for sharing ❤️

    • @cyruschang1904
      @cyruschang1904 14 วันที่ผ่านมา

      @ Thank YOU 🙂

  • @marioalb9726
    @marioalb9726 15 วันที่ผ่านมา +1

    Right triangle BCP
    h/b = 1/2 = h/4 --> h= 2 cm
    Pytagorean theorem:
    s² = b²+h² = 4²+2² = 20cm² (Solved √)

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for sharing ❤️

  • @nenetstree914
    @nenetstree914 15 วันที่ผ่านมา +2

    20

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for sharing ❤️

  • @santiagoarosam430
    @santiagoarosam430 15 วันที่ผ่านมา

    Razón entre catetos =1/2---> PC=4/2=2---> BC=√20---> Área ABCD =20 u².
    Gracias y saludos

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for sharing ❤️

  • @himo3485
    @himo3485 15 วันที่ผ่านมา +1

    MDC∞BPC MD : DC = 1 : 2 PC/4=1/2 PC=2
    BC=√[2²+4²]=√20
    Blue Square area = √20*√20 = 20

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for sharing ❤️

  • @marcgriselhubert3915
    @marcgriselhubert3915 15 วันที่ผ่านมา

    We use an orthonormal center D and first axis (DC) and be c the side length of the square.
    We have C(c; 0) and M(0; c/2). VectorCM(c/2; -c) is colinear to VectorU(-2; 1). The equation of (CM) is: (1).(x - c) - (-2).(y) = 0 or x + 2.y - c = 0
    The distance from B(c; c) to (CM) is abs(c + 2.c - c)/sqrt((2^2) + (1^2)) = (2.c)/(sqrt(5)), and this distance is also 4, so we have (2.c)/(sqrt(5)) = 4,
    so c = 2.sqrt(5) and the area of the square is c^2 = 4.5 = 20.
    (Sorry, I prefer analytic geometry each time it is possible!)

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for the feedback ❤️

  • @PrithwirajSen-nj6qq
    @PrithwirajSen-nj6qq 15 วันที่ผ่านมา

    Sir,
    May I write
    x^2=x√5
    > x =√5(dividing both sides by x)

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Thanks for sharing ❤️

  • @pi5355
    @pi5355 14 วันที่ผ่านมา

    x.2X/2+x.2X/2+4*(5^0.5)x/2=2x.2x
    x^2+x^2+2(5^0.5)x=4x^2
    5^0.5=x
    4x^2=4*5=20

  • @joserubenalcarazmorinigo9540
    @joserubenalcarazmorinigo9540 15 วันที่ผ่านมา +1

    a = lado del cuadrado
    tg A = 1/2
    También cos A = 4/a
    Entonces a = 4 sec A
    a^2 = 16 (sec A)^2 = 16 [1+(tgA)^2]
    a^2=16 [1 + 1/4] = 20

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for sharing ❤️

  • @ChuzzleFriends
    @ChuzzleFriends 14 วันที่ผ่านมา

    Let AM = DM = x. By the Segment Addition Postulate, AD = AM + DM = x + x = 2x.
    By definition of squares, BC = CD = 2x. Use the Pythagorean Theorem on △CDM.
    a² + b² = c²
    x² + (2x)² = (CM)²
    x² + 4x² = (CM)²
    5x² = (CM)²
    CM = √(5x²)
    = x√5
    The root cannot be negative because x must be positive.
    Let α & β be the measures of complementary angles.
    Let m∠CBP = α.
    Then, m∠BCP = β by definition of acute angles in a right triangle.
    Because ∠BCD is a right angle by definition of squares, m∠DCM = α & m∠CMD = β.
    So, △BPC ~ △CDM by SAS. Define proportions.
    BP/CD = BC/CM
    4/(2x) = (2x)/(x√5)
    2x(2x) = 4(x√5)
    4x² = 4x√5
    4x² - 4x√5 = 0
    4x(x - √5) = 0
    4x = 0 or x - √5 = 0
    x = 0 x = √5
    But if x = 0, the diagram would not exist (It would also contradict the statement made earlier about x). So, x = √5.
    The side length of square ABCD is 2x = 2(√5) = 2√5 u.
    Find the area of square ABCD.
    A = s²
    = (2√5)²
    = 2² * (√5)²
    = 4 * 5
    = 20
    So, the area of the blue square is 20 square units.

  • @wackojacko3962
    @wackojacko3962 15 วันที่ผ่านมา +1

    @ 6:47 , 🤔 ... the world needs more ideas and solutions , not fewer! Rejection is a miserable creature. 😊

    • @PreMath
      @PreMath  14 วันที่ผ่านมา +1

      😀
      Thanks for the feedback ❤️

  • @DB-lg5sq
    @DB-lg5sq 15 วันที่ผ่านมา

    شكرا لكم على المجهودات
    يمكن استعمال
    S(ABCD) = 2S(ABM) + S(BMC)
    ........

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for the feedback ❤️🙏

  • @nibiruresearch
    @nibiruresearch 12 วันที่ผ่านมา

    25

  • @EqremKamerri-ch2sp
    @EqremKamerri-ch2sp 15 วันที่ผ่านมา

    √5

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Thanks for sharing ❤️

  • @pedrincalentino6063
    @pedrincalentino6063 3 วันที่ผ่านมา

    🎉Los triángulos PBC y DCM son semejantes, luego los ángulos PCB y CMD son iguales, cuya tangente es 2 A partir de eso, la solució es obvia😂

  • @texitaliano64
    @texitaliano64 12 วันที่ผ่านมา

    Potremmo risolvere una equazione che mette in relazione la superficie del quadrato calcolata utilizzando la formula L*L ovvero considerando L=2x S=4x² e la somma delle superfici dei 3 triangoli DCM ABM MCB, inoltre considerando che MC si può ricavare col teorema di pitagora applicato al triangolo BCM possiamo sviluppare il nostro sistema risolutivo:
    La superficie del triangolo MBA = S1 = 2x*x/2 = x²
    La superficie del triangolo DCM = S2 = 2x*x/2 = x²
    La superficie del triangolo CBM = S3 = MC*4/2 = 2*MC dove MC è la base del triangolo MCB di altezza 4
    con pitagora determino il lato MC
    MC=√(x²+(2x)²) = x√5
    Quindi:
    S=S1+S2+S3
    S=x²+x²+2*x√5
    S=2x²+(2√5)x
    Risolvo il sistema:
    2x²+(2√5)x=4x²
    -2x²+(2√5)x=0
    Questa è una quadratica pertanto:
    x1,2=(-b+/-√(b²-4ac))/2a
    le soluzioni sono x1=0 x2=√5
    scartiamo x1 pertanto la soluzione è x=√5
    sostituendo a S= 4x² otteniamo
    S=4*(√5)²=20
    La superficie del quadrato è 20

  • @sergioaiex3966
    @sergioaiex3966 15 วันที่ผ่านมา +1

    Solution:
    Since M is the midpoint of AD, let's label DM = AM = a
    Therefore AB = BC = CD = DA = 2a
    Applying Pythagorean Theorem in ∆ CDM
    CD² + DM² = CM²
    (2a)² + (a)² = CM²
    CM² = 5a²
    CM = a√5
    ∆ CDM and ∆ BCP are similar by AA, and, like that, we have proportions
    Hence:
    BC/BP = CM/CD
    2a/4 = a√5/2a
    a/2 = √5/2
    a = 2√5/2
    a = √5
    If a = √5, the blue square side length is 2√5
    Blue Square Area = (2√5)²
    Blue Square Area = 20 Square Units ✅

    • @PreMath
      @PreMath  14 วันที่ผ่านมา

      Excellent!
      Thanks for sharing ❤️

  • @puni_arts
    @puni_arts 11 วันที่ผ่านมา

    😂😂median formula pm=4 sind cd since pm90⁰to cd so md=1/2pm so md 2 by applying Pythagarous theorm to triangle pmd we get 20

  • @AssuredSystems
    @AssuredSystems 12 วันที่ผ่านมา

    A right Triangle with one side equal 4, dictates that the other leg is 3 and the hypotunuse is 5. The square has all equal sides of 5 . The area is teh product of the sidea 5 X 5 = 25

    • @shazhu2455
      @shazhu2455 8 วันที่ผ่านมา

      Who was your math teacher😂?

    • @shazhu2455
      @shazhu2455 8 วันที่ผ่านมา

      It’s a 1, 2, sqrt(5) right triangle.

  • @k9slayer
    @k9slayer 15 วันที่ผ่านมา +1

    more magic from the math wizard

    • @PreMath
      @PreMath  15 วันที่ผ่านมา

      Excellent!
      Glad to hear that!
      Thanks for the feedback ❤️

  • @LuisdeBritoCamacho
    @LuisdeBritoCamacho 15 วันที่ผ่านมา +1

    MY RESOLUTION PROPOSAL :
    01) Square Side = 2X lin un
    02) AM = MD = X lin un
    03) (MCD) Angle Tangent = 2X / X = 2
    04) (CMD) Angle Tangent = X / 2X = 1 / 2
    05) Angle (CBP) = Angle (DCM)
    06) Angle (CMD) = Angle (BCM)
    07) Angle (BCM) is a Complemantary Angle with Angle (DCM). DEFINITION : " Two angles are Complementary if the Sum of the Angles is equal to 90º (Ninety Degrees). "
    08) Wich means that Triangle [CDM] is similar to Triangle [CBP]
    09) tan(CBP) = 1 / 2 ; tan(CBP) = PC / 4
    10) 1 / 2 = PC / 4 ; PC = 2 lin un
    11) BC^2 = 4^2 + 2^2 ; BC^2 = (16 + 4) ; BC^2 = 20
    Therefore,
    MY BEST ANSWER IS :
    Blue Square Area equal 20 Square Units.

    • @PreMath
      @PreMath  14 วันที่ผ่านมา +1

      Excellent!
      Thanks for sharing ❤️