Wow! That's a straightforward geometry problem. A great way to end 2024! Happiest New Year 2025, Team Premath! Cheers from The Philippines 🇵🇭🇵🇭🇵🇭🇵🇭🇵🇭🇵🇭
Call it P = 6x and adjust for the missing top right corner later. (x-4)(2x-4) = 96. 2x^2 - 12x + 16 = 96 x^2 - 6x + 8 = 48 x^2 - 6x - 40 = 0 (6+or-sqrt(196))/2 (6+or-14)/2 Delete the negative for x = 10. Therefore, the rectangle would be 60. The removed lines are lengths 16 and 6, but 16 and 6 must also be added. P = 60
This can be done a lot faster, and without needing pen and paper ... the area of the triangle is 1/2(2x-4)(x-4) = 48, giving x=10; now, the perimeter is 2(2x+x) = 60.
Sir I like to focus on the green square. Its side is √16=4 units Then base of the yellow rt 🔺 is (2x -4 ) units and height is (x - 4) units. Hence its area 1/2 *(2x -4)(x -4)=48 >(x -2)^2 -2(x-2)*1+ 1=49 >(x -2-1)^2=7^2 >x -3=7 >x =10 Regarding perimeter of the blue line bounded region we may Shift the middle horizontal line to the topmost horizontal line & the middle vertical line to the rightmost vertical line to complete the 2x X x rectangle and say the Perimeter of the blue line bounded region is 2(2x +x) =6x =6*10=60 units
Length of green square sides = 4. Therefore length of yellow triangle base = 2x - 4. Height of yellow triangle = x - 4. Area of yellow triangle = 1/2 * (2x - 4) * (x - 4). Simplifying = (x - 2) * (x - 4). This = x^2 - 6x + 8. Thus x^2 - 6x + 8 = 48 as given. x^2 - 6x - 40 = 0 (quadratic equation). Factorising, (x - 10) (x + 4) x =10, (-4 ruled out). The base and height of the yellow triangle can be flipped over about its hypotenuse to make a full rectangle. The side lengths of rectangle are x & 2x. Perimeter is 2x + 4x = 6x = 60.
(х-4)(2х-4)/2=48=2(х-4)(х-2)/2=(х-4)(х-2)=6*8. Проверяем х-4=6 ∧ х-2=8. Такое число есть - х=10. Периметр же фигуры равен периметру прямоугольника, т. е. 6х=60 см.
The perimeter is P = 6.x (simple). The side length of the green square is sqrt(16) = 4, so the side lengthes of the yellow triangle are (2.x - 4) and (x - 4) Its double area is (2.x - 4).(x - 4) = 2.(x^2) - 12.x + 16 = 2.48 = 96, so x^2 - 6.x -40 = 0. Deltaprime = 49, then x = 3 - 7 = -4 (rejected, as x > 4) or x = 3 + 7 = 10 Finally, the perimeter is 6.x = 6.10 = 60.
STEP-BY-STEP RESOLUTION PROPOSAL : 01) (X - 4) * (2X - 4) = 96 02) Solution : X = - 4 or X = 10 03) X = 10 04) We can built a Rectangle with Length = 2X and Width = X, with Area = (2X * X) = (20 * 10) = 200 Square Centimeters 05) X + 2X + X + 2X = 4X + 2X = 40 + 20 = 60 cm Therefore, The Blue Polygon Perimeter is equal to 60 Centimeters. Happy New Year!!
Solution: a = side of the square = √16 = 4. The following now applies to the yellow triangle: (2x-4)*(x-4)/2 = 48 |*2 ⟹ 2x²-8x-4x+16 = 96 |-96 ⟹ 2x²-12x-80 = 0 |/2 ⟹ x²-6x-40 = 0 |p-q formula ⟹ x1/2 = 3±√(9+40) = 3±7 ⟹ x1 = 3+7 = 10 and x2 = 3-7 = -4 [useless in geometry] ⟹ Perimeter of the blue polygon = 20+10+10+20 = 60[cm] Happy New Year 2025
Let's face the last challenge in 2024: . .. ... .... ..... First of all we calculate the side length s of the green square: A(square) = s² 16cm² = s² ⇒ s = √(16cm²) = 4cm May h and v be the horizontal and the vertical side length of the yellow right triangle, respectively. From the given diagram we can conclude: A(triangle) = h*v/2 ⇒ v = 2*A(triangle)/h v + s = x h + s = 2x h + s = 2(v + s) h + s = 2v + 2s h − 2v − s = 0 h − 2*[2*A(triangle)/h] − s = 0 h² − s*h − 4*A(triangle) = 0 h² − (4cm)*h − 4*48cm² = 0 h² − (4cm)*h − 192cm² = 0 h = 2cm ± √(4cm² + 192cm²) = 2cm ± √(196cm²) = 2cm ± 14cm Since h>0, the only useful solution is: h = 2cm + 14cm = 16cm ⇒ v = 2*A(triangle)/h = 2*48cm²/(16cm) = 6cm v + s = 6cm + 4cm = 10cm = x h + s = 16cm + 4cm = 20cm = 2x ✅ Now we are able to calculate the perimeter of the blue polygon: P = x + 2x + s + h + v + s = x + 2x + 2x + x = 6x = 60cm Best regards from Germany
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Happy New Year!
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@@PreMath Happy new year everybody!
A = ½bh = ½.(2x-4).(x-4) = 48 cm²
½.(2x²-12x+16) = 48
x² - 6x -40 = 0 --> x= 10cm
P = x+2x+4+(2x-4)+(x-4)+4 = 6x
P = 60 cm ( Solved √ )
True got the same approach since x the height of the longer rectangle
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P=6x
(x-4)(x-2)=48=>x^2-6x=40=>(x-3)=7=>x=10
But you had to find perimeter of blue polygon
@AmirgabYT2185 right u r
Thanks easy❤
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@AmirgabYT2185 Once you have x then the perimeter is trivial. This was another neat little puzzle.
Wow! That's a straightforward geometry problem. A great way to end 2024! Happiest New Year 2025, Team Premath!
Cheers from The Philippines 🇵🇭🇵🇭🇵🇭🇵🇭🇵🇭🇵🇭
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Call it P = 6x and adjust for the missing top right corner later.
(x-4)(2x-4) = 96.
2x^2 - 12x + 16 = 96
x^2 - 6x + 8 = 48
x^2 - 6x - 40 = 0
(6+or-sqrt(196))/2
(6+or-14)/2
Delete the negative for x = 10.
Therefore, the rectangle would be 60.
The removed lines are lengths 16 and 6, but 16 and 6 must also be added.
P = 60
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Have a happy, safe, and prosperous New Year!
This can be done a lot faster, and without needing pen and paper ... the area of the triangle is 1/2(2x-4)(x-4) = 48, giving x=10; now, the perimeter is 2(2x+x) = 60.
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I solved it 🎉🎉🎉
First time
Super!
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16=4^2
(x-4)(2x-4)*1/2=48
2x^2-12x-80=0 x^2-6x-40=0
(x+4)(x-10)=0 x>0 , x=10
Blue Polygon Perimeter = (20+10)*2 = 60cm
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Bom dia Mestre
Obrigado por mais essa aula
Deus lhe abençoe
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Thanks Sir
That’s very nice
Happy new year for all
❤❤❤❤❤❤
Good luck with glades
Thank you!
You're very welcome! 😄
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P=60 cm
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√16=4→ (x-4)(2x-4)=2*48=96→ x=10→ Perímetro del polígono azul =2*x+2*2x= 2*10+2*2*10= 20+40= 60 cm.
Gracias y un saludo cordial.
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Sir I like to focus on the green square. Its side is √16=4 units
Then base of the yellow rt 🔺 is (2x -4 ) units and height is (x - 4) units.
Hence its area
1/2 *(2x -4)(x -4)=48
>(x -2)^2 -2(x-2)*1+ 1=49
>(x -2-1)^2=7^2
>x -3=7
>x =10
Regarding perimeter of the blue line bounded region we may
Shift the middle horizontal line to the topmost horizontal line
& the middle vertical line to the rightmost vertical line to
complete the 2x X x rectangle and say the
Perimeter of the blue line bounded region is
2(2x +x) =6x =6*10=60 units
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@@PreMath Sir let us have a golden year.
Always excellent.
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Length of green square sides = 4.
Therefore length of yellow triangle base = 2x - 4.
Height of yellow triangle = x - 4.
Area of yellow triangle = 1/2 * (2x - 4) * (x - 4).
Simplifying = (x - 2) * (x - 4).
This = x^2 - 6x + 8.
Thus x^2 - 6x + 8 = 48 as given.
x^2 - 6x - 40 = 0 (quadratic equation).
Factorising, (x - 10) (x + 4)
x =10, (-4 ruled out).
The base and height of the yellow triangle can be flipped over about its hypotenuse to make a full rectangle.
The side lengths of rectangle are x & 2x.
Perimeter is 2x + 4x = 6x = 60.
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@@PreMath Happy new year to you, and thanks for keeping our brains exercised and providing excellent education in such a clear manner.
Nice one, many thanks, Sir! (x - 4)(2x - 4) = 96 → x = 10 → perimeter = 60
(x cm - 4 cm)(2x cm - 4 cm)/2 = 48 cm^2
x^2 - 6x - 40 = 0
(x + 4)(x - 10) = 0
x = 10
Perimeter = 2(x + 2x) cm = 60 cm
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@ Thank you. May the New Year bring you and your love ones happiness and success.
(х-4)(2х-4)/2=48=2(х-4)(х-2)/2=(х-4)(х-2)=6*8. Проверяем х-4=6 ∧ х-2=8. Такое число есть - х=10. Периметр же фигуры равен периметру прямоугольника, т. е. 6х=60 см.
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@@PreMath спасибо.
I got this one!
Great!
Have a happy, safe, and prosperous New Year!
60
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4x+8x-16+48*2=2x²
2x²-12x-80=0
x²-6x-40=0
x = 10
*P = 6x = 60*
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Green Square:
A = s²
16 = s²
s = √16 = 4
Yellow Triangle:
A = bh/2
48 = (2x-4)(x-4)/2
(2x-4)(x-4) = 96
2x² - 8x - 4x + 16 - 96 = 0
2x² - 12x - 80 = 0
x² - 6x - 40 = 0
(x-10)(x+4) = 0
x = 10 | -x = -4- ❌ x > 0
P = x + 2x + 4 + (2x-4) + (x-4) + 4
P = x + 2x + 2x + x
[ P = 6x = 60cm ]
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The perimeter is P = 6.x (simple). The side length of the green square is sqrt(16) = 4, so the side lengthes of the yellow triangle are (2.x - 4) and (x - 4)
Its double area is (2.x - 4).(x - 4) = 2.(x^2) - 12.x + 16 = 2.48 = 96, so x^2 - 6.x -40 = 0. Deltaprime = 49, then x = 3 - 7 = -4 (rejected, as x > 4) or x = 3 + 7 = 10
Finally, the perimeter is 6.x = 6.10 = 60.
Please edit to show a correct sol
@@PrithwirajSen-nj6qq Done, sorry.
@marcgriselhubert3915
Need not to be sorry. This typographical error is natural. Thanks a lot.
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h = x - 4
b = 2x-4
(x-4)(2x-4) = 48
---------
2
2x²-4x-8x+16 = 48
-----------
2
x²-6x+8 = 48
x²-6x =40
Completando cuadrados:
x²-6x+9 = 40+9
x²-6x+9 = 49
(x-3)² = 49
x-3 = ±7
X1 = 10
X2 = -4 ....imposible
X=10
Perimetro :
x+2x+x+2x = 60
Happy New Year 🎉
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1/2 x (x-4) x (2x-4) = 48
x = 10
p = 4 + 10 + 20 + 4 + 16 + 6 = 60
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STEP-BY-STEP RESOLUTION PROPOSAL :
01) (X - 4) * (2X - 4) = 96
02) Solution : X = - 4 or X = 10
03) X = 10
04) We can built a Rectangle with Length = 2X and Width = X, with Area = (2X * X) = (20 * 10) = 200 Square Centimeters
05) X + 2X + X + 2X = 4X + 2X = 40 + 20 = 60 cm
Therefore,
The Blue Polygon Perimeter is equal to 60 Centimeters.
Happy New Year!!
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perm=60 cm
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No thinkin outside the box today... Just careful reasoning and good conceptual understanding of what the hell is goin on! 😊
😮😅😊
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Solution:
a = side of the square = √16 = 4.
The following now applies to the yellow triangle:
(2x-4)*(x-4)/2 = 48 |*2 ⟹
2x²-8x-4x+16 = 96 |-96 ⟹
2x²-12x-80 = 0 |/2 ⟹
x²-6x-40 = 0 |p-q formula ⟹
x1/2 = 3±√(9+40) = 3±7 ⟹
x1 = 3+7 = 10 and x2 = 3-7 = -4 [useless in geometry] ⟹
Perimeter of the blue polygon = 20+10+10+20 = 60[cm]
Happy New Year 2025
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Sir,solve this X^X^X= 16, prove x=2
x^x^x = 16
To prove: x=2
Now,
x^x^x = 16
or, x^x^x = 2^4
or, x^x^x = 2^2^2
Hence, x=2 PROVED.
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Solution:
A = s²
16 = s²
s = 4
2x = a + 4 ... ¹
x = b + 4 ... ²
Substituting ² in ¹
2 (b + 4) = a + 4
2b + 8 = a + 4
a = 2b + 4
Yellow Triangle Area = ½ a b ... ³
48 = ½ (2b + 4) b
48 = ½ (2b² + 4b)
96 = 2b² + 4b (÷2)
48 = b² + 2b
b² + 2b - 48 = 0
b = [-2 ± √(4 + 4 . 1 . 48)]/2
b = (-2 ± 14)/2
b' = 6 Accepted
b" = - 8 Rejected
48 = ½ a . 6
a = 16
x = 6 + 4
x = 10
Perimeter = x + 2x + 4 + a + b + 4
Perimeter = 3x + a + b + 8
Perimeter = 3 . (10) + 16 + 6 + 8
Perimeter = 30 + 30
Perimeter = 60 cm ✅
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12×4=48+8=56
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Let's face the last challenge in 2024:
.
..
...
....
.....
First of all we calculate the side length s of the green square:
A(square) = s²
16cm² = s²
⇒ s = √(16cm²) = 4cm
May h and v be the horizontal and the vertical side length of the yellow right triangle, respectively. From the given diagram we can conclude:
A(triangle) = h*v/2 ⇒ v = 2*A(triangle)/h
v + s = x
h + s = 2x
h + s = 2(v + s)
h + s = 2v + 2s
h − 2v − s = 0
h − 2*[2*A(triangle)/h] − s = 0
h² − s*h − 4*A(triangle) = 0
h² − (4cm)*h − 4*48cm² = 0
h² − (4cm)*h − 192cm² = 0
h = 2cm ± √(4cm² + 192cm²) = 2cm ± √(196cm²) = 2cm ± 14cm
Since h>0, the only useful solution is:
h = 2cm + 14cm = 16cm
⇒ v = 2*A(triangle)/h = 2*48cm²/(16cm) = 6cm
v + s = 6cm + 4cm = 10cm = x
h + s = 16cm + 4cm = 20cm = 2x ✅
Now we are able to calculate the perimeter of the blue polygon:
P = x + 2x + s + h + v + s = x + 2x + 2x + x = 6x = 60cm
Best regards from Germany
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Have a happy, safe, and prosperous New Year!
You are doing a great job! Can you help me with something unrelated? I have a SafePal wallet with TRC20 USDT in it and I have my recovery phrase: (pride pole obtain together second when future mask review nature potato bulb). How do I transfer them to Binance?
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Have a happy, safe, and prosperous New Year!
60
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